The World in Motion: A Clear Guide to Foundational Physics

Book Introduction

Move beyond formula memorization. This visual guide explains motion, forces, energy, heat, waves, sound, and circuits through everyday examples, worked problems, investigation tools, and an original story—for teens and returning adults.

Contents

1. Measuring the World

2. Position and Velocity

3. Acceleration and Changing Motion

4. Finding and Drawing Forces

5. Using Newton’s Laws

6. Motion Under Gravity

7. Work and Mechanical Energy

8. Heat, Temperature, and Internal Energy

9. What Waves Carry

10. Reading Sound as Physics

11. Current and Electric Circuits

12. Electrical Power and Safe Use

13. Energy and Society

14. Using Physics in an Investigation

15. End Matter and Review

16. Closing Note

17. Short Story: One Second Before the Raindrop Stopped

18. Four-Panel Epilogue: Beyond the Falling Drop

Foundational Physics and the World in Motion: A Briefing Document

Executive Summary

Physics is not a discipline confined to laboratories; it is a systematic framework for understanding the "world in motion" that defines everyday life. The core philosophy of this briefing is that physics is a durable skill involving the identification of relationships and conditions rather than a mere hunt for formulas. This document synthesizes the foundational principles of motion, energy, waves, and electricity, emphasizing a consistent methodology of visualization, systematic analysis, and honest measurement.

Critical takeaways include:

• Relationship over Memorization: Formulas are compact statements about relationships that only hold under specific conditions.

• Interactions and Conservation: Physical events are driven by interactions (forces) and governed by conservation laws (energy, charge).

• Measurement Integrity: Scientific investigation is an "argument about how much confidence a number deserves," requiring honesty regarding uncertainty and outliers.

• Societal Context: Energy use is a chain of conversions where efficiency and lifecycle impacts must be evaluated through multiple lenses (reliability, safety, economics, and environment).

I. The Methodology of Physical Analysis

The study of physics requires a consistent working method to move between words, diagrams, graphs, and equations.

The Seven-Step Solution Method

1. Restate the Event: Reduce the situation to its physical core (e.g., constant speed, insulated mixing).

2. Choose the System: Define what is inside the boundary of interest.

3. Choose Reference/Direction: State the positive direction and zero points for position or potential energy.

4. Draw: Utilize free-body diagrams, circuits, or energy chains.

5. List Quantities: Identify knowns and unknowns with SI units.

6. Select a Law: Choose a principle (e.g., Ohm’s law, conservation of energy) based on valid assumptions.

7. Check: Verify units, magnitude, and significant figures.

II. Measurement and Motion (Kinematics)

Physics begins by turning impressions into comparable quantities.

Fundamentals of Measurement

• SI Units: The shared language of science, including meters (m), kilograms (kg), seconds (s), amperes (A), and kelvins (K).

• Precision and Limits: Measurements have finite resolution. Significant figures must be used to avoid claiming false precision.

• Scalars vs. Vectors:

  • Scalars: Magnitude only (mass, time, temperature, energy, speed).

  • Vectors: Magnitude and direction (displacement, velocity, acceleration, force).

Describing Change

• Velocity: The rate of change of position. Average velocity describes an interval, while instantaneous velocity describes a specific moment.

• Acceleration: The rate at which velocity changes. A negative acceleration does not always mean slowing down; speed increases when velocity and acceleration point in the same direction.

• Graphs as Maps:

  • Position–Time Slope: Represents velocity.

  • Velocity–Time Slope: Represents acceleration.

  • Velocity–Time Area: Represents displacement.

III. Forces and Newton’s Laws (Dynamics)

Forces are interactions between two objects; they cause changes in velocity rather than maintaining it.

Core Principles

1. Inertia (First Law): Objects resist changes in motion. Net force of zero means constant velocity or rest.

2. Acceleration (Second Law): F_{net} = ma. Acceleration is determined by the vector sum of all forces and the object's mass.

3. Action–Reaction (Third Law): Forces always exist in pairs of equal magnitude and opposite direction acting on different objects.

Specific Interactions

• Weight vs. Mass: Mass is a measure of inertia (kg); weight is the gravitational force (W = mg) measured in Newtons.

• Friction: Opposes relative sliding. Static friction adjusts up to a maximum (f_{max} = \mu_sN), while kinetic friction acts during motion.

• Normal Force: A contact force perpendicular to a surface, resulting from equilibrium.

IV. Energy, Work, and Thermal Physics

Energy is a conserved quantity that reaches us through conversion chains.

Work and Mechanical Energy

• Work: Energy transferred when a force acts through a displacement (W = Fs \cos \theta).

• Conservation of Mechanical Energy: In the absence of nonconservative forces (like friction), the sum of kinetic (K) and potential (U) energy remains constant.

• Efficiency: The ratio of useful output energy to total input energy.

Thermal Principles

• Heat vs. Temperature: Temperature relates to microscopic thermal motion; heat is the process of energy transfer due to temperature differences.

• Internal Energy: The sum of microscopic kinetic and potential energies of particles.

• Phase Changes: During melting or boiling, temperature remains constant because energy (latent heat) is used to change particle arrangements rather than their speed.

• Modes of Transfer: Conduction (matter interactions), Convection (fluid motion), and Radiation (electromagnetic waves).

V. Waves and Sound

Waves transport energy and information through a propagating disturbance without the medium itself traveling.

Wave Behavior

• Quantities: Speed (v), frequency (f), wavelength (\lambda), and period (T). Relationship: v = f\lambda.

• Types:

  • Transverse: Oscillation perpendicular to propagation (e.g., rope waves).

  • Longitudinal: Oscillation parallel to propagation (e.g., sound in air).

• Superposition: When waves overlap, their displacements add. This leads to interference (constructive or destructive).

• Resonance: Occurs when a driving force matches a system’s natural frequency, producing large amplitudes.

VI. Electricity and Circuits

Electrical systems operate on the conservation of charge and energy.

Circuit Fundamentals

• Current (I): The rate of charge flow (I = Q/t).

• Voltage (V): Energy transferred per unit charge (V = W/Q).

• Ohm’s Law: V = RI. Resistance (R) is a property of the conductor's material and geometry.

Circuit Configurations

Feature

Series Circuit

Parallel Circuit

Current

Same current through each component

Current splits at junctions

Voltage

Total voltage is sum of drops

Same voltage across each branch

Resistance

R = R_1 + R_2 + \dots

1/R = 1/R_1 + 1/R_2 + \dots

VII. Energy and Society

Societal energy decisions require a lifecycle perspective beyond simple operation.

Comparison Lenses

1. Reliability: Delivery of energy when needed.

2. Environmental Impact: Lifecycle effects on climate, water, and ecosystems.

3. Safety: Likelihood and consequence of failures.

4. Economics: Costs of construction, fuel, and end-of-life management.

5. Resources/Fairness: Distribution of benefits and risks.

Nuclear Energy and Radiation

• Units:

  • Becquerel (Bq): Nuclear decays per second.

  • Gray (Gy): Energy absorbed per kilogram.

  • Sievert (Sv): Dose adjusted for biological effect.

• Safety Principles: Reduce time, increase distance, and utilize appropriate shielding.

VIII. The Scientific Investigation

Investigation turns curiosity into measurable questions through controlled comparison.

• Variables:

  • Independent: Deliberately changed.

  • Dependent: Measured in response.

  • Controlled: Kept constant to ensure a fair test.

• Data Integrity: Unexpected results or outliers (the "strange point") should not be deleted automatically. They often indicate uncontrolled variables, perspective errors (like parallax), or flaws in the initial model.

• Causation: Correlation does not automatically imply causation; causal claims require repeatable, controlled comparison and a plausible mechanism.

"Measurement is not a contest to guess the approved number. It is an argument about how much confidence a number deserves."

The World in Motion: Foundational Physics Study Guide

This study guide provides a structured review of the core principles of foundational physics as outlined in the source text. It covers measurement, kinematics, dynamics, energy, thermodynamics, wave mechanics, electricity, and the societal impacts of energy systems.

Part 1: Short-Answer Quiz

Instructions: Answer each of the following questions in two to three sentences based on the principles described in the source context.

1. Distinguish between a scalar and a vector quantity, providing an example of each.

2. How does displacement differ from distance in the context of a round trip?

3. Explain why negative acceleration does not necessarily mean an object is slowing down.

4. What is the difference between mass and weight, and how do they change in different gravitational environments?

5. Define Newton's Third Law and explain why action-reaction pairs do not cancel each other out.

6. Under what specific conditions is mechanical energy conserved within a system?

7. Describe the difference between temperature and heat according to the microscopic behavior of particles.

8. What determines the speed of a mechanical wave, and how does wavelength change if the frequency of the source increases?

9. Explain the difference between series and parallel circuits regarding current and voltage.

10. In a scientific investigation, what is the role of an independent variable compared to a controlled variable?

Part 2: Quiz Answer Key

1. Scalars vs. Vectors: A scalar is a quantity that has magnitude but no direction, such as mass or temperature. A vector requires both magnitude and direction to be fully described, such as displacement or force, and is often represented by an arrow.

2. Displacement vs. Distance: Distance is a scalar that accumulates along the entire path traveled, whereas displacement is a vector representing the change in position from the start to the end. In a round trip, the distance is the total path length (e.g., 400 m), but the displacement is zero because the final and initial positions are identical.

3. Negative Acceleration: The sign of acceleration indicates its direction relative to a chosen positive origin, not necessarily a change in speed. Speed only decreases if velocity and acceleration point in opposite directions; if both are negative, the object is actually speeding up in the negative direction.

4. Mass vs. Weight: Mass measures an object's inertia in kilograms and remains constant regardless of location. Weight is the gravitational force acting on that mass (W = mg); it changes based on the local gravitational field strength, meaning an object weighs less on the Moon than on Earth despite having the same mass.

5. Newton’s Third Law: This law states that if object A exerts a force on object B, object B simultaneously exerts a force of equal magnitude and opposite direction on object A. These forces do not cancel because they act on different objects rather than a single body in equilibrium.

6. Conservation of Mechanical Energy: Mechanical energy, the sum of kinetic and potential energy, is conserved only when no nonconservative forces (like friction or air resistance) do work on the system. If friction acts, mechanical energy is converted into other forms, such as internal energy or sound, though total energy remains conserved.

7. Temperature vs. Heat: Temperature is a measure of the microscopic thermal motion of particles, whereas heat is the energy transferred between objects specifically because of a temperature difference. An object contains internal energy, but "heat" only refers to the transfer process itself until thermal equilibrium is reached.

8. Wave Speed and Frequency: The speed of a mechanical wave is determined primarily by the properties of the medium, such as tension in a rope or the temperature of air. Because wave speed is constant for a given medium, increasing the source frequency results in a shorter wavelength (λ = v/f).

9. Series vs. Parallel Circuits: In a series circuit, there is only one path for charge, so the current is the same through all components while the source voltage is split among them. In a parallel circuit, components are connected across the same two nodes, meaning they all share the same voltage while the total current splits among the different branches.

10. Variables in Investigation: The independent variable is the factor that is deliberately changed by the researcher to observe an effect. In contrast, controlled variables are factors kept constant to ensure that any observed changes in the dependent variable are solely due to the manipulation of the independent variable.

Part 3: Essay Questions

Instructions: Use the principles and data points provided in the source context to develop comprehensive responses to the following prompts.

1. The Geometry of Motion: Analyze how graphs function as "compressed maps of change." Specifically, explain the physical significance of the slope and the area under the curve for both position–time and velocity–time graphs, and describe how linearization (such as plotting y against t^2) helps scientists test physical models.

2. The Dual Nature of Friction: Discuss friction as both a resistive force and a necessary interaction. Cover the distinctions between static and kinetic friction, the mathematical models used to calculate them, and provide examples of how friction is essential for everyday activities like walking or driving.

3. Thermodynamics and Phase Changes: Explain the relationship between internal energy, specific heat capacity, and latent heat. Describe what happens to energy and temperature during a phase change (like melting ice) and explain why a bathtub of warm water might have more internal energy than a small cup of boiling water.

4. Energy and Society: Compare and contrast at least three different primary energy sources (e.g., thermal, solar, nuclear, or hydro) using the "five lenses" of comparison provided in the text: reliability, environmental impact, safety, economics, and resources/fairness.

5. The Ethics of Measurement: Drawing on the short story "One Second Before the Raindrop Stopped," discuss the importance of honesty and methodological rigor in scientific investigations. Address how "bad data" or unexpected results can lead to a deeper understanding of systematic errors, such as perspective or calibration issues.

Part 4: Comprehensive Glossary

Term

Definition / Formula

Absolute Temperature

Temperature measured in Kelvins (K); T = t[°C] + 273.15.

Acceleration

The rate at which velocity changes; a = \Delta v / \Delta t, measured in m/s^2.

Amplitude (A)

The maximum displacement of a wave from its equilibrium position.

Becquerel (Bq)

The SI unit of nuclear activity, representing one nuclear decay per second.

Current (I)

The rate at which electric charge passes a cross-section; I = \Delta Q / \Delta t, measured in Amperes (A).

Diffraction

The spreading of waves after passing through an opening or bending around obstacles.

Efficiency (η)

The ratio of useful output energy to total input energy, often expressed as a percentage.

Force

An interaction that can change an object’s velocity or deform it; measured in Newtons (N).

Frequency (f)

The number of oscillations or cycles per second; measured in Hertz (Hz).

Gray (Gy)

The SI unit for absorbed radiation dose, representing energy absorbed per kilogram.

Inertia

The tendency of an object to resist changes in its state of motion; measured by mass.

Joule (J)

The SI unit of work and energy; 1 J = 1 N·m.

Kinetic Energy (K)

The energy of motion; K = 1/2 mv^2.

Latent Heat (L)

The energy required per kilogram to change the phase of a substance without changing its temperature.

Net Force

The vector sum of all forces acting on a single object.

Normal Force (N)

A contact force exerted perpendicular to a surface that supports an object.

Ohm’s Law

The principle that current is proportional to voltage for some conductors; V = RI.

Power (P)

The rate of doing work or transferring energy; P = W / t, measured in Watts (W).

Potential Energy (U)

Stored energy based on position (gravitational: mgh) or deformation (elastic: 1/2 kx^2).

Resistivity (ρ)

A property of a material that determines its electrical resistance based on geometry: R = ρL/S.

Sievert (Sv)

The SI unit for radiation dose adjusted for biological effect.

Specific Heat (c)

The energy required to raise the temperature of 1 kg of a substance by 1 Kelvin.

Standing Wave

A wave pattern produced by the interference of two waves of the same frequency traveling in opposite directions; characterized by nodes and antinodes.

Voltage (V)

The electric potential energy transferred per unit charge between two points; measured in Volts (V).

Wavelength (λ)

The distance between two neighboring points in the same phase on a wave (e.g., crest to crest).

Work (W)

Energy transferred when a force acts through a displacement; W = Fs \cos \theta.

The World in Motion
A Clear Guide to Foundational Physics: Motion, Energy, Waves, Electricity, and Everyday Life

Magic Carpet Ride High school

Author: Magic Carpet Ride Promotion
Publisher: Magic Carpet Ride Publishing

Introduction

When a train leaves a station, your body seems to lean backward. A hot drink left on a desk gradually cools. A voice from a speaker reaches you as a changing pressure pattern in the air. When a phone charges, electrical energy is converted and stored in another form. Physics is not confined to laboratories. It is already built into an ordinary day.

Foundational physics is sometimes taught as a hunt for the correct formula. That approach works only until a problem changes its wording. The more durable skill is learning to identify what matters in a situation, represent it clearly, and connect the relevant quantities. A formula is not a magic command. It is a compact statement about a relationship and the conditions under which that relationship holds.

This book is written for high school learners and adults starting again from the beginning. It follows the core scope of Japan’s current high school Physics Basics curriculum while adapting the explanations for an international English-reading audience. The chapters cover measurement, motion, forces, energy, heat, waves, sound, circuits, electrical power, energy systems, and scientific investigation.

The working method is consistent throughout: picture the event, choose a system and a positive direction, draw a diagram, list known and unknown quantities, select a law with valid assumptions, calculate with units, and check whether the result makes sense. Examples explain why an equation applies, not only how to substitute numbers.

As you learn to move among words, diagrams, graphs, and equations, physics stops looking like a wall of symbols. It becomes a way to read change. The bicycle outside your window, the light over your desk, and the echo in a hallway all begin to tell a more detailed story.

The world is always in motion. Let us learn how to listen to what that motion says.

Chapter 1 — Measuring the World

Physics begins by making comparisons possible

“The runner was fast” is an impression. “The runner covered 100 m in 14.0 s” is a measurement that other people can check and compare. Physics turns observations into quantities that can be tested.

A physical quantity combines a number with a unit. Length, time, mass, temperature, and electric current are examples. Writing that a desk is “1.2 long” is incomplete. A value of 1.2 m describes a desk; 1.2 cm describes something very different. Units are part of the meaning, not decoration added after a calculation.

The International System of Units, or SI, provides a shared language. Important base units in this book include the meter (m), kilogram (kg), second (s), ampere (A), and kelvin (K). Derived units combine them. A newton (N) measures force, a joule (J) measures energy, and a watt (W) measures energy transferred per second.

The unit m/s can be read literally: meters traveled in each second. To convert 36 km/h to m/s, convert the distance and time separately:

36 km/h = 36,000 m / 3,600 s = 10 m/s

Following the units makes the conversion understandable instead of turning it into a memorized trick.

Prefixes and scientific notation

Physics deals with quantities ranging from power-station output to microscopic time intervals. Prefixes keep the numbers manageable.

Prefix Symbol Factor Example giga G 10^9 1 GW = 10^9 W mega M 10^6 1 MJ = 10^6 J kilo k 10^3 1 km = 10^3 m centi c 10^-2 1 cm = 10^-2 m milli m 10^-3 1 mm = 10^-3 m micro μ 10^-6 1 μs = 10^-6 s

Capitalization matters: M means mega, while m used as a prefix means milli. The SI base unit of mass is the kilogram, so 250 g must be written as 0.250 kg before it is used in many equations.

Scientific notation writes a quantity as a number from 1 up to, but not including, 10 multiplied by a power of ten. Thus 0.00042 s becomes 4.2 × 10^-4 s, and 3,600,000 J becomes 3.6 × 10^6 J. The exponent immediately shows the order of magnitude.

Measurements have limits

Repeated measurements rarely agree perfectly. A scale has finite resolution. An observer may read at a slight angle. Temperature may change the apparatus. These are not reasons to pretend the data are exact; they are reasons to report what the measurement can support.

A ruler marked every millimeter cannot justify a result carried to one-millionth of a millimeter. In introductory work, significant figures are used to avoid claiming false precision. The number 12.4 cm has three significant figures. The number 0.0320 kg also has three: the leading zeros locate the decimal point, while the final zero communicates precision.

For multiplication and division, a common rule is to round the final result to the same number of significant figures as the least precise input. If a rectangle measures 12.4 cm by 3.2 cm, the calculator gives 39.68 cm². Because 3.2 has two significant figures, the area is reported as 4.0 × 10^1 cm².

Keep extra digits during intermediate calculations and round at the end. Rounding each line can accumulate unnecessary error.

Scalars and vectors

A quantity that has magnitude but no direction is a scalar. Mass, elapsed time, temperature, energy, and speed are scalars. A quantity that needs both magnitude and direction is a vector. Displacement, velocity, acceleration, and force are vectors.

Suppose you walk 300 m east, then 100 m west. The total distance traveled is 400 m. Your displacement from the starting point is 200 m east. Distance is a scalar accumulated along the path; displacement is the change in position.

For motion along a line, choose a positive direction and use signed numbers. If east is positive, an eastward displacement is positive and a westward displacement is negative. A minus sign means “opposite the chosen positive direction,” not “bad” or “slower.”

In two dimensions, vectors are drawn as arrows. Arrow length represents magnitude and arrow direction represents direction. Components allow one vector to be separated into perpendicular directions, such as horizontal and vertical.

Graphs are compressed maps of change

A table preserves exact values, but a graph reveals patterns. Before interpreting a graph, read both axes, their units, the scale, and the origin. Then ask what the slope and area mean.

On a position–time graph, slope represents velocity. On a velocity–time graph, slope represents acceleration, while signed area between the graph and the time axis represents displacement. These relationships connect the geometry of a graph with physical definitions.

Do not assume that a high point on any graph means “fast.” A high value on a position–time graph means the object is far from the origin. Speed is indicated by how steeply position changes with time.

Checking an equation with units

Dimensional checking catches many mistakes. Distance found from speed and time should have units

(m/s) × s = m.

If your calculation produces m/s² when the question asks for a distance, something is wrong. Units cannot confirm a missing factor such as 1/2 or determine direction, but they are a powerful first check.

Worked example — reaction distance

A cyclist moves at a steady 6.0 m/s. It takes 0.70 s to begin braking after noticing a hazard. How far does the bicycle travel during that reaction time?

The speed is constant, so

distance = speed × time

= 6.0 m/s × 0.70 s

= 4.2 m.

The unit check gives (m/s) × s = m. Even a short decision time matters because the bicycle continues moving before braking begins.

Check your understanding

1. Convert 72 km/h to m/s.

2. Write 0.0035 A in scientific notation.

3. If north is positive, write a velocity of 4.0 m/s south as a signed value.

4. Find the distance traveled at 8.0 m/s for 2.5 s.

Answers

1. 20 m/s.

2. 3.5 × 10^-3 A.

3. -4.0 m/s.

4. 20 m.

Chapter 2 — Position and Velocity

Position needs a reference

To describe motion, choose an origin and a positive direction. If the center of a hallway is x = 0 and east is positive, a point 5 m east is x = +5 m and a point 3 m west is x = -3 m.

When an object moves from x1 to x2, its displacement is

Δx = x2 - x1.

The symbol Δ means “change in,” calculated as final value minus initial value. Moving from -2 m to +4 m gives Δx = 4 - (-2) = +6 m.

Distance and displacement are not interchangeable. Completing one lap of a 400 m track gives a distance of 400 m but a displacement of 0 because the final and initial positions are the same.

Average velocity describes the whole interval

Average velocity over a time interval is

v = Δx/Δt = (x2 - x1)/(t2 - t1).

Because displacement is a vector, average velocity includes direction. If east is positive and you move 20 m east in 5.0 s, the average velocity is +4.0 m/s. If you travel out and return to the starting point, the average velocity for the complete trip is zero even though the average speed is not.

Speed is the magnitude of velocity. Average speed is total distance divided by total time. It cannot be negative.

Instantaneous velocity

A vehicle’s speedometer does not report the average since the journey began. It estimates the speed at that moment. Instantaneous velocity can be understood as average velocity over an increasingly short time interval.

On a position–time graph, the slope of a line joining two points is average velocity for that interval. As the interval shrinks, the line approaches the tangent at one point. The tangent’s slope is instantaneous velocity.

Position–time and velocity–time graphs comparing constant motion and accelerated motion.

Figure 1 — Connecting position–time and velocity–time graphs

A straight line rising on a position–time graph indicates constant positive velocity. A horizontal line indicates rest. A falling line indicates negative velocity. A curve indicates that the slope, and therefore the velocity, changes with time.

Constant-velocity motion

An object moving along a straight line with constant velocity follows

x = x0 + vt,

where x0 is its position at t = 0. The shorter form x = vt is valid only when x0 = 0.

An object starting at x0 = -10 m and moving at +3.0 m/s has position

x = -10 + (3.0)(5.0) = +5.0 m

after 5.0 s. It moved 15 m, but its final position is +5.0 m relative to the chosen origin.

Relative velocity

Two trains traveling side by side at nearly the same velocity can appear almost stationary to one another. The velocity of B relative to A is

vBA = vB - vA.

If east is positive, A moves at +20 m/s and B at +25 m/s, then B moves at +5 m/s relative to A. A moves at -5 m/s relative to B.

The same equation works for opposite directions. If A is +20 m/s and B is -15 m/s, then B’s velocity relative to A is -35 m/s. Using signs is more reliable than memorizing separate rules for “same direction” and “opposite direction.”

Meeting points

Two objects meet when their positions are equal at the same time. On a position–time graph, the meeting event is the intersection of their lines.

Object A starts at x = 0 and moves at +2.0 m/s. Object B starts at x = 30 m and moves at -3.0 m/s. Their positions are

xA = 2.0t

xB = 30 - 3.0t.

Setting them equal gives

2.0t = 30 - 3.0t,

so t = 6.0 s. The meeting position is x = 12 m. Equivalently, the 30 m gap closes at a relative speed of 5.0 m/s.

Area under a velocity–time graph

For constant velocity, displacement is Δx = vΔt. On a velocity–time graph, v is the height and Δt is the width, so their product is the rectangular area.

When velocity changes, divide the interval into narrow strips. Adding their signed areas gives displacement. Area above the time axis is positive; area below is negative. The units confirm the interpretation:

(m/s) × s = m.

If an object moves at +3.0 m/s for 4.0 s and then at -1.0 m/s for 2.0 s, its displacement is

(3.0)(4.0) + (-1.0)(2.0) = +10 m.

Its distance is 12 m + 2 m = 14 m.

Measuring velocity from video

A video divides motion into images separated by known time intervals. At 60 frames per second, adjacent frames are 1/60 s apart. If a scale in the image converts pixel position into real distance, positions can be measured and average velocities calculated.

Good geometry matters. A camera aimed at an angle creates perspective error. Place the scale in the same plane as the motion, keep the camera fixed and square to that plane, and use a clearly defined tracking point on the object. Repeat the measurement and report scatter instead of hiding it.

Worked example — an out-and-back trip

A person walks 120 m east in 30 s, then 40 m west in 10 s. Take east as positive.

Total displacement = +120 - 40 = +80 m.

Total time = 40 s.

Average velocity = 80/40 = +2.0 m/s.

Total distance = 120 + 40 = 160 m.

Average speed = 160/40 = 4.0 m/s.

Check your understanding

1. Find the displacement from x = -4.0 m to x = +11 m.

2. A runner completes a 200 m lap in 40 s and returns to the start. Find average velocity and average speed.

3. An object starts at x0 = 8.0 m and moves at -2.0 m/s. Find its position after 6.0 s.

4. A car moving east at 15 m/s observes a bicycle moving east at 10 m/s. Find the bicycle’s velocity relative to the car.

Answers

1. +15 m.

2. Average velocity 0 m/s; average speed 5.0 m/s.

3. -4.0 m.

4. -5.0 m/s.

Chapter 3 — Acceleration and Changing Motion

The rate at which velocity changes

A train leaving a station gains speed, cruises, and then slows for the next stop. Acceleration describes how quickly velocity changes, including changes of direction.

Average acceleration is

a = Δv/Δt = (v2 - v1)/(t2 - t1).

Its SI unit is m/s². An acceleration of 1 m/s² means that velocity changes by 1 m/s during each second.

If velocity changes from +4.0 m/s to +10 m/s in 2.0 s,

a = (10 - 4.0)/2.0 = +3.0 m/s².

Negative acceleration does not always mean slowing down

The sign of acceleration indicates direction. Speed increases when velocity and acceleration point in the same direction and decreases when they point in opposite directions.

Velocity Acceleration What happens to speed? positive positive increases positive negative decreases negative positive decreases negative negative increases

If velocity changes from -2 m/s to -6 m/s, the signed value becomes more negative, but speed increases from 2 m/s to 6 m/s. Always compare directions.

Three equations for constant acceleration

For motion along a straight line with constant acceleration, initial position x0, initial velocity v0, and elapsed time t,

v = v0 + at

x = x0 + v0t + 1/2 at²

v² - v0² = 2a(x - x0).

The first relates velocity and time. The second gives position. The third is useful when time is unknown or unnecessary. Before substitution, choose a positive direction and assign signs to v0 and a.

These equations are not valid for every changing motion. Their defining assumption is constant acceleration over the interval.

Where the factor 1/2 comes from

With constant acceleration, a velocity–time graph is a straight line. Its slope is acceleration. Displacement is the area under the line.

The area consists of a rectangle v0t plus a triangle with base t and height v - v0 = at. Therefore

displacement = v0t + 1/2(at)t

= v0t + 1/2 at².

The factor 1/2 is the geometry of a triangle, not an arbitrary number to memorize.

Another useful result is that, for constant acceleration, average velocity is

vavg = (v0 + v)/2.

Then displacement is vavg t.

Worked example — accelerating bicycle

A bicycle moves at +2.0 m/s and accelerates at +1.5 m/s² for 4.0 s.

Final velocity:

v = 2.0 + (1.5)(4.0) = 8.0 m/s.

Displacement:

Δx = (2.0)(4.0) + 1/2(1.5)(4.0²)

= 8.0 + 12

= 20 m.

The average-velocity method gives ((2.0 + 8.0)/2) × 4.0 = 20 m as a check.

Stopping distance

An object slowing under constant acceleration stops when v = 0. If v0 = +12 m/s and a = -3.0 m/s²,

0² - 12² = 2(-3.0)Δx,

so Δx = 24 m.

If the initial speed doubles while the braking acceleration stays the same, stopping distance becomes four times as large because it depends on v0². Real traffic stopping distance also includes the distance traveled before the brakes begin to act.

Piecewise graphs and models

When the slope of a velocity–time graph changes, acceleration changes. Analyze each interval separately. A horizontal segment means zero acceleration, not zero velocity.

Experimental points will not usually fall on a perfect line. Surface irregularities, air resistance, timing resolution, and reading error cause scatter. A best-fit line estimates the overall acceleration. The equation is a model that preserves the dominant relationship while neglecting smaller effects.

Check your understanding

1. Velocity changes from +3.0 m/s to +18 m/s in 5.0 s. Find the average acceleration.

2. An object has v0 = +10 m/s and a = -2.0 m/s². How long does it take to stop?

3. An object starts from rest and accelerates at +2.0 m/s² for 3.0 s. Find its displacement.

4. If v = -4.0 m/s and a = -1.0 m/s², is speed increasing or decreasing?

Answers

1. +3.0 m/s².

2. 5.0 s.

3. 9.0 m.

4. Increasing, because velocity and acceleration point in the same direction.

Chapter 4 — Finding and Drawing Forces

A force is an interaction

A force can change an object’s velocity or deform the object. A hand pushes a cart, Earth pulls a ball, a table supports a book, and a stretched spring pulls toward its natural length. In every case, identify the interaction by saying what exerts the force and what receives it.

Force is a vector measured in newtons. An arrow shows its direction and magnitude. A common misconception is that a moving object must have a force pointing along its motion. In the absence of a net force, an object can keep moving at constant velocity. Force causes a change in velocity; it is not required to maintain velocity.

Mass and weight

Mass measures inertia and is expressed in kilograms. Weight is the gravitational force on an object and is measured in newtons. Near Earth’s surface,

W = mg,

where g is approximately 9.8 m/s².

A 2.0 kg object weighs 19.6 N near Earth’s surface. Its mass would remain 2.0 kg on the Moon, but its weight would be smaller because the local gravitational field is weaker.

Normal force

A book on a table does not fall through it because the table exerts a contact force perpendicular to the surface. This is the normal force.

On a horizontal table, a book at rest with no other vertical forces has a normal force equal in magnitude to its weight. That is a result of equilibrium, not a universal formula. Push down on the book and the table’s normal force becomes larger than mg. On an incline, the normal force balances the component of weight perpendicular to the surface.

Friction

Friction acts along a contact surface and opposes relative sliding or the tendency to slide. Static friction adjusts from zero up to a maximum:

fmax = μsN.

If you push with 5 N and an object remains at rest, static friction is 5 N in the opposite direction, provided 5 N is below the maximum. It is not automatically equal to μsN.

Once surfaces slide, kinetic friction is often modeled as

fk = μkN.

Friction is not merely a loss. Walking requires static friction from the ground. Tires grip, brakes work, and pencils leave marks because of friction.

Spring force

Within its elastic range, a spring’s restoring force is proportional to its extension or compression x:

F = kx.

The spring constant k is measured in N/m. A larger k means a stiffer spring. Including direction, the relation is often written F = -kx, where the minus sign indicates that the spring force points opposite the displacement from equilibrium.

If a mass hangs at rest from a vertical spring, the upward spring force balances the downward weight, so kx = mg.

Adding and resolving forces

The single force with the same effect as all forces combined is the net force. Along one line, assign signs and add. In two dimensions, resolve forces into perpendicular components.

If a force F makes an angle θ above the horizontal,

Fx = F cos θ

Fy = F sin θ.

Choosing axes along and perpendicular to an incline often simplifies the problem. Coordinates are tools; choose the orientation that makes the physics easiest to see.

A free-body diagram showing weight, normal force, applied force, and friction on a block.

Figure 2 — Isolating one object in a free-body diagram

Equilibrium

Forces are in equilibrium when their vector sum is zero. The object may be at rest or moving with constant velocity.

Two forces alone balance only if they are equal in magnitude, opposite in direction, and act along the same line on the same object. For three or more forces, the vector sum must still be zero.

Balanced forces and action–reaction pairs are different. Balanced forces act on one object. An action–reaction pair acts on two different objects. The book’s weight and the table’s normal force balance on the book. The book pushing the table and the table pushing the book form an action–reaction pair.

How to draw a free-body diagram

1. Choose the object of interest.

2. Draw that object as a point or simple shape.

3. Identify every object in contact with it.

4. Add noncontact forces such as gravity.

5. Draw only forces acting on the chosen object.

6. Choose coordinate directions.

Do not add an arrow simply because the object is moving. Do not draw a force that the object exerts on something else.

Worked example — a block on a horizontal surface

A 3.0 kg block is pushed 12 N to the right. Kinetic friction is 5.0 N to the left. The horizontal net force is

12 - 5.0 = 7.0 N to the right.

Vertically, the 29.4 N weight and 29.4 N normal force balance. Horizontally, the nonzero net force means the velocity will change. The next chapter connects that force to acceleration.

Check your understanding

1. Find the weight of a 2.5 kg object using g = 9.8 m/s².

2. An object remains at rest when pushed right with 6.0 N. The maximum static friction is 10 N. Find the actual friction force.

3. A spring with k = 200 N/m is stretched 0.030 m. Find the force magnitude.

4. Forces of 8.0 N right and 3.0 N left act on one object. Find the net force.

Answers

1. 24.5 N.

2. 6.0 N left.

3. 6.0 N.

4. 5.0 N right.

Chapter 5 — Using Newton’s Laws

Inertia

When a train starts suddenly, your feet accelerate with the floor while the rest of your body tends to remain in its original state of motion. When the train brakes, your body tends to continue forward.

Newton’s first law states that if the net force on an object is zero, an object at rest remains at rest and an object in motion continues at constant velocity along a straight line. The tendency to resist changes in velocity is inertia.

Everyday objects usually slow because friction and air resistance create a net force. The observation that they stop does not mean that force is required to keep them moving.

Newton’s second law

The acceleration of an object is determined by net force and mass:

Fnet = ma.

Acceleration points in the direction of the net force. One newton is the force that gives a 1 kg mass an acceleration of 1 m/s².

A reliable procedure is:

1. Select the object or system.

2. Choose positive directions.

3. Draw all forces acting on it.

4. Add force components in each direction.

5. Write Fnet = ma for each direction.

6. Solve with SI units and check the result.

The F in F = ma is not automatically one named force. It is the vector sum of all forces on the chosen object.

Worked example — friction and acceleration

A 4.0 kg box is pulled 18 N to the right while kinetic friction acts 6.0 N left. The net horizontal force is +12 N, so

12 = 4.0a,

giving a = +3.0 m/s².

If the box starts from rest and moves for 3.0 s under the same forces, its speed becomes 9.0 m/s and its displacement is 13.5 m. Force analysis gives acceleration; constant-acceleration equations then describe the motion.

Mass as resistance to acceleration

With the same net force, a larger mass has a smaller acceleration. A 10 N net force gives a 2.0 kg object an acceleration of 5.0 m/s², but a 5.0 kg object only 2.0 m/s².

This is why mass is a measure of inertia. It also explains why greater force is needed to give a loaded cart the same acceleration as an empty cart.

Newton’s third law

If object A exerts a force on object B, object B simultaneously exerts a force of equal magnitude and opposite direction on object A:

FAB = -FBA.

A swimmer pushes water backward; the water pushes the swimmer forward. A rocket pushes exhaust backward; the exhaust pushes the rocket forward. Your foot pushes the ground backward; the ground pushes you forward.

These forces do not cancel because they act on different objects. Always attach an object name to each force.

Treating several objects as a system

Two blocks of 2.0 kg and 3.0 kg are connected by a light string on a frictionless surface and pulled by 10 N. Treating both blocks as one system gives

a = 10/(2.0 + 3.0) = 2.0 m/s².

Tension is internal to the two-block system, so the action–reaction pair cancels in the system equation. To find the tension, isolate one block. If tension is the only horizontal force on the 2.0 kg block,

T = ma = (2.0)(2.0) = 4.0 N.

Use the whole system for common acceleration and a subsystem for internal forces.

Apparent weight in an elevator

A scale responds to the normal force, not directly to gravity. For a person of mass m in an elevator, taking upward as positive,

N - mg = ma.

When the elevator accelerates upward, N > mg and the person feels heavier. When it accelerates downward, N < mg. At constant velocity, a = 0 and N = mg. Direction of motion is not enough; direction of acceleration matters.

A 50 kg person accelerating upward at 1.2 m/s² experiences

N = 50(9.8 + 1.2) = 550 N.

Motion on an incline

For an incline at angle θ, resolve weight into components:

• parallel to the slope: mg sin θ

• perpendicular to the slope: mg cos θ.

On a smooth slope, N = mg cos θ and

mg sin θ = ma,

so a = g sin θ. Mass cancels.

With kinetic friction coefficient μk, the friction magnitude is μkmg cos θ. Taking downhill as positive,

mg sin θ - μkmg cos θ = ma.

Air resistance and terminal speed

Real falling objects experience drag that generally increases with speed. At first, weight is greater than drag, so the object accelerates downward. If drag grows until it equals weight, the net force becomes zero and the object continues at a constant terminal speed.

Raindrops and parachutes cannot be modeled well without drag. A dense compact object falling a short distance may be approximated by neglecting drag. The right model depends on shape, scale, speed, and required accuracy.

Check your understanding

1. A 2.0 kg object has forces of 9.0 N right and 3.0 N left. Find its acceleration.

2. If net force is unchanged and mass triples, what happens to acceleration?

3. In an elevator moving upward at constant velocity, how does scale force compare with weight?

4. What is the reaction to the table pushing upward on a book?

Answers

1. 3.0 m/s² right.

2. It becomes one-third as large.

3. They are equal in magnitude.

4. The book pushes downward on the table.

Chapter 6 — Motion Under Gravity

A useful idealization

Near Earth’s surface, an object in free fall experiences weight mg. If air resistance is neglected,

mg = ma,

so a = g. The mass cancels, meaning objects of different mass have the same gravitational acceleration under the same conditions.

A flat sheet of paper falls more slowly than a metal ball because drag is important. Crumple the paper and the difference becomes smaller. In a vacuum, both share the same acceleration.

Free fall from rest

Choose downward as positive, set the release point to y = 0, and let initial velocity be zero. Then

v = gt

y = 1/2 gt²

v² = 2gy.

After 1.0 s, speed is 9.8 m/s and distance fallen is 4.9 m. After 2.0 s, speed is 19.6 m/s and distance is 19.6 m. Doubling time doubles speed but multiplies distance by four.

For a drop from 20 m,

20 = 1/2(9.8)t²,

so t is about 2.0 s and impact speed is about 20 m/s in the no-drag model.

Thrown downward

With downward positive and initial downward speed v0,

v = v0 + gt

y = v0t + 1/2 gt².

Free fall from rest is the special case v0 = 0. When a quadratic equation produces a negative time, that solution may describe an extrapolated part of the mathematical path before the chosen start time; it is not part of the observed event.

Thrown upward

Take upward as positive. Gravity points downward, so a = -g:

v = v0 - gt

y = v0t - 1/2 gt².

At the highest point, velocity is momentarily zero, but acceleration remains -g. For v0 = 19.6 m/s, the highest point is reached after 2.0 s and lies 19.6 m above the release point. With no drag, the object returns to its original height after 4.0 s with the same speed in the opposite direction.

Either upward or downward can be positive. Consistency matters more than the choice.

Horizontal launch

An object launched horizontally combines two independent motions that share the same time:

• horizontal motion at constant velocity

• vertical free fall.

Taking downward as positive,

x = v0t

y = 1/2 gt².

Objects launched horizontally at different speeds from the same height land at the same time in the no-drag model, but travel different horizontal distances.

From a height of 19.6 m, the fall time is 2.0 s. A horizontal launch speed of 10 m/s gives a range of 20 m. Just before landing, the horizontal velocity is still 10 m/s while the vertical component is 19.6 m/s downward.

Graphs and linearization

For downward-positive free fall, the velocity–time graph is a straight line through the origin with slope g. The triangular area under it is 1/2 gt², the displacement.

A graph of y against t is curved. Because y is proportional to t², plotting y against t² should produce a straight line with slope g/2. This transformation, called linearization, makes a relationship easier to test and its parameter easier to estimate.

A linearized free-fall graph of distance y against time squared, with slope corresponding to half the gravitational acceleration.

Figure 3 — Linearizing free-fall data

Measurement and safety

Do not drop objects from high places or above people, animals, or fragile property. Use a light object, a low controlled height, and a protected landing area. In school, follow the teacher’s procedure.

For video analysis, secure the camera, place the scale in the same plane as the fall, and use frame count for timing. A short safe drop filmed at high frame rate can reveal the motion without a hazardous setup.

Worked example — timing a distant impact

Suppose a small stone is dropped into a deep shaft and the impact is heard 3.0 s later. Treating 3.0 s as fall time gives about 44 m, but the measurement also includes the time for sound to travel back upward.

If the depth is h and sound speed is 340 m/s,

fall time = √(2h/g)

sound travel time = h/340.

Their sum equals 3.0 s. Solving numerically gives a depth of about 40 m. The example shows why a measured interval must be separated into the processes it contains.

Check your understanding

1. Find the speed after 1.5 s of free fall from rest, using g = 9.8 m/s².

2. If free-fall time triples, by what factor does distance increase?

3. An object is thrown upward at 14.7 m/s. How long until it reaches the top?

4. For a horizontal launch from unchanged height, what happens to fall time and horizontal range when horizontal speed doubles?

Answers

1. 14.7 m/s.

2. Nine times.

3. 1.5 s.

4. Fall time is unchanged; range doubles.

Chapter 7 — Work and Mechanical Energy

Work in physics

In physics, work describes energy transferred when a force acts through a displacement. For a constant force F parallel to displacement s,

W = Fs.

The unit is the joule: 1 J = 1 N·m. If force and displacement form an angle θ,

W = Fs cos θ.

Work is positive when the force component points along the displacement, negative when it points opposite, and zero when force and displacement are perpendicular.

Holding a bag while walking horizontally can be tiring, but the upward supporting force does zero mechanical work on the bag because its displacement is horizontal. Your body still converts chemical energy internally. “Work” always refers to a specified force acting on a specified system.

Positive and negative work

Positive work tends to increase an object’s kinetic energy. Braking and friction often do negative work.

A box pulled 5.0 m by a 10 N horizontal force receives +50 J of work from that force. If friction is 4.0 N opposite the motion, friction does -20 J. Weight and normal force do zero work during horizontal displacement. Net work is +30 J.

For a changing force, work equals signed area under a force–position graph. The area unit, N·m, confirms that it represents joules.

Power

Power is the rate of doing work or transferring energy:

P = W/t.

One watt is one joule per second. The watt rating on an appliance describes how rapidly it converts energy, not the total energy used.

When a force F acts along constant velocity v,

P = Fv.

Moving the same load to the same height requires the same ideal work whether done quickly or slowly, but doing it quickly requires greater power.

Kinetic energy

The kinetic energy of an object of mass m and speed v is

K = 1/2 mv².

It is proportional to mass and to the square of speed. Doubling speed multiplies kinetic energy by four. Kinetic energy is a scalar and never negative.

Using F = ma and v² - v0² = 2as,

Fs = mas = 1/2 m(v² - v0²).

Therefore,

net work = change in kinetic energy

Wnet = ΔK.

This is the work–energy theorem.

Gravitational potential energy

Near Earth’s surface, gravitational potential energy relative to a chosen zero height is

U = mgh.

The zero level is arbitrary. Only changes in potential energy affect predictions, so keep the same reference throughout a problem.

When an object moves downward, gravitational potential energy decreases and gravity does positive work:

Wgravity = -ΔU.

Elastic potential energy

For an ideal spring displaced by x within its elastic range,

U = 1/2 kx².

Spring force rises from zero to kx as the spring is stretched, so the average force over the stretch is kx/2. Doubling the deformation stores four times as much energy.

Conservation of mechanical energy

Mechanical energy is the sum of kinetic and potential energy:

E = K + U.

If only conservative forces such as gravity and ideal spring force do work, and no energy crosses the chosen system boundary,

K1 + U1 = K2 + U2.

For an object dropped from height h,

mgh = 1/2 mv²,

so v = √(2gh). Mass cancels.

Energy bars showing gravitational potential energy decreasing while kinetic energy increases, with total mechanical energy constant.

Figure 4 — Mechanical energy during a fall

When friction acts

Mechanical energy may decrease when friction acts, but total energy has not vanished. It becomes internal energy of the surfaces, sound, deformation, and other forms.

A 2.0 kg object begins with 98 J of gravitational potential energy and reaches the bottom with 64 J of kinetic energy. The missing 34 J of mechanical energy has been transferred mainly into internal energy.

Mechanical energy is conserved only under stated conditions; total energy is conserved while changing form.

Efficiency

Efficiency compares useful output with input:

η = useful output energy / input energy.

Multiply by 100% for a percentage. If 500 J enters a device and 350 J becomes the intended output, the efficiency is 70%. An efficiency above 100% signals a mistaken system boundary, unit error, or missing input.

Worked example — speed on a track

A vehicle moves at 5.0 m/s at height 20 m, then descends without friction to height 5.0 m. Conservation of mechanical energy gives

1/2 m(5.0²) + mg(20) = 1/2 mv² + mg(5.0).

Mass cancels, leaving

v² = 5.0² + 2(9.8)(15),

so v ≈ 18 m/s.

Check your understanding

1. A 12 N force moves an object 3.0 m in the force direction. Find the work.

2. Find the kinetic energy of a 2.0 kg object moving at 4.0 m/s.

3. Find the increase in potential energy when a 3.0 kg object is lifted 2.0 m.

4. A device turns 600 J of 800 J input into useful output. Find its efficiency.

Answers

1. 36 J.

2. 16 J.

3. 58.8 J.

4. 75%.

Chapter 8 — Heat, Temperature, and Internal Energy

Temperature is not heat

A small cup of hot water and a bathtub of warm water cannot be compared by temperature alone. The amount and type of matter also affect stored internal energy.

Temperature is related to the microscopic thermal motion of particles. Heat is energy transferred because of a temperature difference. An object contains internal energy; heat is a transfer process.

When hot and cold objects touch, energy moves from the hotter object to the colder one until they reach thermal equilibrium. At equilibrium, particles still move, but there is no net macroscopic heat transfer between the objects.

Celsius and absolute temperature

Celsius temperature t and absolute temperature T are related by

T = t + 273.15.

Thus 0°C is 273.15 K and 20°C is 293.15 K. A temperature difference of 1°C is the same size as a difference of 1 K.

Statements such as “twice as hot” require care. Doubling 20°C to 40°C does not double absolute temperature. Ratios of thermal temperature should use kelvins.

Internal energy

Particles translate, rotate, vibrate, and interact. Their microscopic kinetic and potential energies make up a system’s internal energy. Heating usually increases internal energy. During melting or boiling, internal energy can increase even while temperature stays nearly constant because the arrangement and interactions of particles change.

The kinetic energy of an entire moving cup is not internal energy. Separating macroscopic motion from microscopic energy is important.

Heat capacity and specific heat

Heat capacity C is the energy required to raise an entire object’s temperature by 1 K. Specific heat capacity c is the energy required to raise 1 kg of a substance by 1 K. Without a phase change,

Q = mcΔT.

Water has a relatively large specific heat, about 4.2 × 10³ J/(kg·K). It changes temperature slowly for a given energy input, which makes it useful for cooling and thermal storage.

To heat 0.50 kg of water from 20°C to 60°C,

Q = (0.50)(4.2 × 10³)(40)

= 8.4 × 10⁴ J.

Real heaters must supply more because the container and surroundings also gain energy.

Calorimetry and thermal equilibrium

If a system is well insulated, energy lost by warmer objects equals energy gained by cooler objects:

total heat released = total heat absorbed.

Mix 0.10 kg of water at 80°C with 0.20 kg at 20°C. Ignoring the container, let the final temperature be T:

0.10c(80 - T) = 0.20c(T - 20).

Solving gives T = 40°C. It is not the simple average because the cooler sample has twice the mass.

Phase changes and latent heat

Melting, freezing, vaporization, and condensation are phase changes. During a phase change at nearly constant pressure, supplied energy may change particle arrangement rather than temperature.

The energy required is

Q = mL,

where L is the specific latent heat. Heating ice at 0°C to water above 0°C requires separate stages: melt the ice using mL, then warm the water using mcΔT.

Evaporation cools skin because the escaping molecules take energy from the liquid and surroundings. Refrigerators and air conditioners move energy using phase changes of a working fluid, together with compression and expansion.

Three modes of thermal transfer

Conduction transfers energy through microscopic interactions within matter. Metals conduct effectively in part because mobile electrons carry energy.

Convection transfers energy through bulk motion of a fluid. Warm, less dense fluid may rise while cooler fluid sinks.

Radiation transfers energy by electromagnetic waves and can cross a vacuum. Energy from the Sun reaches Earth by radiation.

Insulation works by limiting one or more of these pathways. A vacuum flask reduces conduction and convection across the evacuated gap, while reflective surfaces reduce radiation.

Work and internal energy

A bicycle pump can become warm when air is compressed quickly. Work done on the gas raises its internal energy. If Q is heat received by a system and W is work done on it,

ΔU = Q + W.

Some textbooks define W as work done by the gas and write a minus sign instead. Always identify the sign convention. The physical energy accounting is the same.

Power and heating efficiency

A 1.0 kW heater operating for 120 s supplies

E = Pt = (1.0 × 10³)(120) = 1.2 × 10⁵ J.

If 9.0 × 10⁴ J reaches the water, the heating efficiency is 75%. A lid, appropriate quantity, and better insulation reduce unwanted transfer to the environment.

Check your understanding

1. Convert 25°C to kelvins.

2. A 2.0 kg object has c = 500 J/(kg·K). Find the energy needed for a 6.0 K increase.

3. How much energy does a 100 W device transfer in 30 s?

4. Why can temperature remain nearly constant while ice melts?

Answers

1. About 298 K.

2. 6.0 × 10³ J.

3. 3.0 × 10³ J.

4. Energy changes the particle arrangement and interactions during the phase change.

Chapter 9 — What Waves Carry

A disturbance travels; the medium does not travel with it

Drop a pebble into a pond and ripples move outward. The water does not flow all the way to shore with each crest. Local regions move mainly around equilibrium while passing the disturbance to neighboring regions. A wave transports energy and information through a propagating disturbance.

The material through which a mechanical wave travels is the medium. Water carries water waves; air carries ordinary sound. Electromagnetic waves, including light and radio, can also travel through a vacuum.

It helps to keep two views separate: one point in the medium oscillates, while the pattern of that oscillation travels through space.

Transverse and longitudinal waves

In a transverse wave, the medium oscillates perpendicular to the direction of propagation. A wave on a stretched rope is a common example.

In a longitudinal wave, the medium oscillates parallel to the direction of propagation. Compression pulses in a spring and sound in air are examples. Compressions and rarefactions travel even though individual particles move back and forth over small distances.

A longitudinal wave can be drawn as a graph with peaks and valleys if the vertical axis represents pressure, density, or longitudinal displacement. The graph does not imply that air molecules are moving vertically.

Wave quantities

Amplitude A is the maximum displacement from equilibrium. Larger amplitude generally means more energy is carried.

Wavelength λ is the distance between neighboring points in the same phase, such as crest to crest.

Period T is the time for one complete oscillation. Frequency f is the number of oscillations per second, measured in hertz:

f = 1/T.

A wave travels one wavelength during one period, so

v = fλ = λ/T.

A sinusoidal wave labeled with amplitude, wavelength, period, direction of propagation, and oscillation direction.

Figure 5 — Basic wave quantities

What determines wave speed

Under fixed conditions, wave speed is determined mainly by the medium. On a rope, tension and mass per unit length matter. For water waves, depth can matter. Sound speed depends on the medium and temperature.

If the source frequency changes while speed in the medium stays approximately constant, wavelength changes according to λ = v/f. Shaking a rope faster creates shorter wavelengths; it does not necessarily make the disturbance travel faster.

At a boundary between media, speed and wavelength may change. Frequency remains fixed by the source, so a slower speed produces a shorter wavelength.

Phase

Phase describes the stage of an oscillation. Points with the same displacement and motion direction are in phase. A crest and a trough are half a cycle out of phase.

In a traveling periodic wave, points one wavelength apart are in phase; points half a wavelength apart are in opposite phase. At one location, waiting half a period also advances the phase by half a cycle.

Phase becomes essential when waves overlap.

Superposition and interference

When waves share a region, their instantaneous displacements add. This is the principle of superposition. In an ideal linear medium, the waves continue afterward with their identities intact.

In-phase waves reinforce one another, producing constructive interference. Equal waves in opposite phase can cancel at a point, producing destructive interference.

Active noise control creates a sound designed to be approximately opposite in phase to unwanted sound at the listener’s location. It cannot cancel every frequency at every point because propagation paths, sensors, and processing delay matter.

Reflection

At a boundary, part of a wave may return. A pulse reflecting from a fixed end of a rope is inverted because the end must remain at zero displacement. A pulse reflecting from an ideal free end is not inverted.

For many wave types, reflection angle equals incidence angle when angles are measured from the normal to the boundary.

Standing waves

Two waves with the same frequency and amplitude traveling in opposite directions can create a standing wave. The pattern does not move along the medium.

Points that never move are nodes. Points with maximum oscillation are antinodes. Adjacent nodes are λ/2 apart, as are adjacent antinodes. A node and its nearest antinode are λ/4 apart.

A string fixed at both ends must have nodes at both ends. If its length is L, the fundamental mode has L = λ/2. Higher modes fit an integer number of half-wavelengths into L.

Diffraction

Waves spread after passing through an opening and bend around obstacles. Diffraction is strongest when the opening or obstacle size is comparable to the wavelength.

Low-frequency sound has a longer wavelength and often bends around doorways and walls more noticeably than high-frequency sound. Visible light has a much shorter wavelength than ordinary objects, so everyday shadows are relatively sharp.

Following a waveform in time

A wave moving right at speed v shifts a distance vΔt to the right after time Δt without changing shape in an ideal medium. To determine whether a particular point is moving up or down, sketch the waveform a short time later and compare the point’s displacement.

This small-shift method is more dependable than memorizing a rule detached from the drawing.

Worked example — finding wavelength

A 12 Hz wave travels at 3.0 m/s:

λ = v/f = 3.0/12 = 0.25 m.

If frequency doubles to 24 Hz while the medium and speed remain unchanged, wavelength halves to 0.125 m.

Check your understanding

1. Find the frequency of a wave with period 0.040 s.

2. Find the speed of a wave with λ = 0.80 m and f = 5.0 Hz.

3. Adjacent nodes in a standing wave are 0.30 m apart. Find λ.

4. A wave enters a slower medium with unchanged frequency. What happens to wavelength?

Answers

1. 25 Hz.

2. 4.0 m/s.

3. 0.60 m.

4. It becomes shorter.

Chapter 10 — Reading Sound as Physics

Sound is a traveling pressure change

A speaker cone moving forward compresses nearby air. Moving backward creates a rarefaction. This pressure pattern travels and vibrates the eardrum. Sound in air is longitudinal.

Sound also travels through liquids and solids. It cannot travel through a vacuum because there is no material medium to carry the mechanical disturbance.

Sound speed in air changes with temperature. Near ordinary conditions, an estimate is

v ≈ 331.5 + 0.6t m/s,

where t is Celsius temperature. At 20°C, this gives about 344 m/s. Humidity and atmospheric conditions produce smaller corrections.

Pitch, loudness, and timbre

Pitch is associated mainly with frequency. Higher frequency is heard as a higher pitch. Human hearing varies, but a commonly stated range is roughly 20 Hz to 20 kHz. Frequencies above the audible range are called ultrasound.

Loudness is related to wave amplitude and intensity, but perception also depends on frequency and the listener. Timbre distinguishes sounds with similar pitch and loudness. It depends on the mixture of frequency components and on how the sound begins and decays.

Waveforms and spectra

A microphone converts sound into an electrical signal that can be plotted against time. A pure tone resembles a sine wave. Speech and musical instruments contain many frequencies and create more complex waveforms.

A frequency spectrum shows how strongly different frequencies are present. The time waveform answers “when and how did the signal change?” The spectrum answers “which frequencies are included?”

Phone apps can display spectra, but the device microphone is not a calibrated laboratory instrument. Avoid excessive sound levels and do not record private conversations without permission.

Echoes and reverberation

If a reflected sound returns after time Δt, distance d to the reflecting surface is

d = vΔt/2.

The factor 1/2 accounts for the outbound and return paths. A reflection heard 0.40 s after the sound, with v = 340 m/s, comes from a surface 68 m away.

In a room, many reflections overlap. The persistence of sound is reverberation. Some reverberation can enrich music, while too much reduces speech clarity. Acoustic design balances reflecting, scattering, and absorbing surfaces for the room’s purpose.

Natural modes of a string

A string fixed at both ends has nodes at both ends. If n half-wavelengths fit into length L,

L = nλn/2

and

fn = nv/(2L).

The n = 1 mode is the fundamental. Shortening the string raises its frequency. Increasing tension raises wave speed and pitch; increasing mass per unit length tends to lower them.

Resonance in air columns

In an ideal pipe open at both ends, both ends are displacement antinodes. The fundamental has L = λ/2:

f1 = v/(2L).

An ideal pipe closed at one end has a node at the closed end and an antinode at the open end. Its fundamental has L = λ/4:

f1 = v/(4L).

Ideal closed pipes support odd multiples of the fundamental. Real pipes require an end correction because oscillating air extends slightly beyond an open end.

Resonance

Every oscillating system has natural frequencies. A periodic driving force near a natural frequency can produce a large amplitude. This is resonance.

A tuning fork sounds louder on a resonant box. A swing gains amplitude when pushed at the right timing. Musical instruments use resonance deliberately. Buildings and machines must often avoid or damp dangerous resonance.

Beats

Two close frequencies produce alternating constructive and destructive interference heard as beats. The beat frequency is

fbeat = |f1 - f2|.

Tones at 440 Hz and 443 Hz produce three beats per second. Musicians can tune by reducing the beat rate toward zero.

Doppler effect

An approaching siren sounds higher and a receding siren lower. Relative motion between source and observer changes the observed frequency. When a source approaches, wavefronts in front are closer together, so wavelength is shorter. When it recedes, wavelength is longer.

The source’s own oscillation frequency has not necessarily changed. Any formula must clearly state the directions of source, observer, and medium.

Sound intensity and hearing safety

Sound intensity is power per unit area, measured in W/m². For an ideal point source spreading uniformly, doubling distance spreads the same power over four times the area, reducing intensity to one-fourth.

Sound level in decibels uses a logarithmic scale, so differences are not read as simple percentages. Long exposure to high levels can damage hearing. Keep personal audio at a moderate setting, take breaks, and stop if discomfort or ringing persists. Seek qualified health advice for continuing symptoms.

Worked example — an open pipe

An open pipe 0.85 m long has sound speed 340 m/s. Its fundamental frequency is

f1 = 340/[2(0.85)] = 200 Hz.

The next ideal modes are 400 Hz and 600 Hz.

Check your understanding

1. Estimate sound speed at 20°C.

2. An echo returns after 0.60 s. Find the wall distance using 340 m/s.

3. Find the beat frequency of 256 Hz and 260 Hz.

4. Find the fundamental of a 0.50 m open pipe at 340 m/s.

Answers

1. About 344 m/s.

2. 102 m.

3. 4 Hz.

4. 340 Hz.

Chapter 11 — Current and Electric Circuits

Electric charge

Static shocks occur when electrons move between materials and create an imbalance of charge. Charge can be positive or negative. Like charges repel and opposite charges attract. The SI unit is the coulomb (C). One electron carries approximately -1.60 × 10^-19 C.

An object becomes positively charged mainly by losing electrons and negatively charged by gaining them. In an isolated system, total charge is conserved.

Conductors allow charge carriers to move readily. In metals, mobile electrons carry charge. Insulators restrict motion, but no everyday material is a perfect insulator under all voltages, humidity levels, and conditions.

Current

Current is the rate at which charge passes a cross section:

I = ΔQ/Δt.

One ampere is one coulomb per second. If 12 C passes in 3.0 s, average current is 4.0 A. If 0.20 A flows for 30 s, the transferred charge is 6.0 C.

Conventional current is defined in the direction positive charge would move. In a metal, electrons move in the opposite direction.

Voltage

Voltage is energy transferred per unit charge between two points:

V = W/Q.

One volt is one joule per coulomb. Current flows; voltage does not “flow.” Voltage is a difference between points.

A battery raises the electric potential energy of charge. When a circuit is closed, charge moves through components and transfers energy. A real battery has internal resistance, so terminal voltage may fall under heavy load.

Ohm’s law

For some conductors at nearly constant temperature, current is proportional to voltage:

V = RI.

Resistance R is measured in ohms. A 6.0 Ω resistor across 12 V carries 2.0 A.

Not every component is ohmic. A filament lamp heats as current rises, changing its resistance. A diode behaves very differently in opposite directions. A straight V–I graph through the origin indicates ohmic behavior over the measured range.

Resistivity and geometry

For a uniform wire,

R = ρL/S,

where ρ is resistivity, L is length, and S is cross-sectional area. A longer wire has greater resistance; a thicker wire has smaller resistance.

Metal resistance generally rises with temperature because lattice vibrations interfere more strongly with electron motion. Some semiconductor resistance falls as temperature rises because more charge carriers become available. Temperature sensors can use this behavior.

Series circuits

Components in a single path are in series. In steady state, the same current passes through each:

I = I1 = I2 = ….

Source voltage equals the sum of component voltage drops:

V = V1 + V2 + ….

Therefore the equivalent resistance is

R = R1 + R2 + ….

A 4.0 Ω and 8.0 Ω resistor in series across 12 V have equivalent resistance 12 Ω and current 1.0 A. Their voltage drops are 4.0 V and 8.0 V.

Parallel circuits

Components connected across the same two nodes are in parallel. Each branch has the same voltage:

V = V1 = V2 = ….

Current splits at a junction:

I = I1 + I2 + ….

The equivalent resistance satisfies

1/R = 1/R1 + 1/R2 + ….

Parallel equivalent resistance is smaller than any branch resistance because more current paths are available.

A 6.0 Ω and 3.0 Ω resistor in parallel across 6.0 V carry 1.0 A and 2.0 A. Total current is 3.0 A, so equivalent resistance is 2.0 Ω.

A comparison of series and parallel circuits, showing common current in series and common voltage in parallel.

Figure 6 — Series and parallel circuits

Meters

An ammeter is connected in series so the measured current passes through it. An ideal ammeter has negligible resistance. Connecting an ammeter directly across a source can create a dangerous short circuit.

A voltmeter is connected in parallel across the two points of interest. An ideal voltmeter has very large resistance and draws negligible current.

Begin with an appropriate high range, check polarity, and change connections only with the source disconnected. Never connect classroom equipment or a homemade circuit to household mains.

Conservation behind circuit rules

At a junction, total current entering equals total current leaving because charge does not accumulate indefinitely. Around a complete loop, energy supplied per coulomb equals energy transferred per coulomb in components. Series and parallel rules are practical consequences of charge and energy conservation.

Worked example — a series–parallel circuit

A 3.0 Ω resistor R1 is in series with two 6.0 Ω resistors R2 and R3 in parallel. The supply is 12 V.

The parallel pair has equivalent resistance 3.0 Ω. Total resistance is 6.0 Ω, so total current is 2.0 A. R1 drops 6.0 V, leaving 6.0 V across each parallel branch. Each 6.0 Ω branch carries 1.0 A, and the branch currents add to 2.0 A.

Check your understanding

1. How much charge passes when 0.50 A flows for 20 s?

2. What voltage corresponds to 9.0 J per coulomb?

3. Find the series equivalent of 5.0 Ω and 7.0 Ω.

4. Find the parallel equivalent of 6.0 Ω and 3.0 Ω.

Answers

1. 10 C.

2. 9.0 V.

3. 12 Ω.

4. 2.0 Ω.

Chapter 12 — Electrical Power and Safe Use

The rate of electrical energy conversion

Electrical devices convert energy into light, heat, motion, sound, and information processing. If charge Q moves through voltage V, energy transfer is W = VQ. Because Q = It,

W = VIt

and electrical power is

P = VI.

For an ohmic resistor,

P = I²R = V²/R.

The statement “larger resistance produces more heating” is incomplete. At fixed current, I²R increases with R. At fixed voltage, V²/R decreases with R. Always state what is held constant.

Joule heating

Resistance converts electrical energy into internal energy:

Q = VIt = I²Rt = V²t/R.

Heaters use this deliberately. In power lines and cables it is usually a loss. Because heating grows with current squared, undersized or damaged cords can overheat. Follow the current and power rating, keep cords uncoiled when required by the manufacturer, and replace damaged equipment.

Energy use and kilowatt-hours

Electrical energy is E = Pt. The SI unit is the joule, but utility energy is commonly measured in kilowatt-hours:

1 kWh = 3.6 × 10⁶ J.

An 800 W device used for 0.50 h consumes

0.800 kW × 0.50 h = 0.40 kWh.

Electricity tariffs vary by location, contract, and date. Use the current official tariff when estimating cost rather than treating an example price as permanent.

Direct and alternating current

Direct current maintains one direction. Batteries and many electronic circuits use DC. Alternating current changes magnitude and direction periodically. Utility supplies use AC because voltage can be transformed efficiently.

The stated household AC voltage is an rms value: it produces the same average heating in a resistor as that DC voltage. For an ideal sine wave, peak voltage is √2 times rms voltage.

Japan uses both 50 Hz and 60 Hz utility regions. Many modern devices support both, but always check the rating label.

Magnetic effects and motors

An electric current creates a magnetic field around a conductor. Coiling the wire reinforces the field, and an iron core can create an electromagnet. Relays, speakers, and motors use controlled magnetic forces.

A current-carrying conductor in a magnetic field experiences a force. A motor converts electrical energy into mechanical rotation. Reversing current reverses the force direction if the magnetic field is unchanged.

Electromagnetic induction and generation

A changing magnetic flux through a coil induces a voltage. Moving a magnet into or out of a coil or rotating a coil in a magnetic field can create current in a closed circuit.

The induced current acts in a direction that opposes the change producing it. Faster change, more turns, and stronger magnetic field generally increase induced voltage.

A generator converts mechanical energy into electrical energy. It does not create energy from nothing. A hand generator becomes harder to turn under load because induced effects oppose the motion that produces the current.

Transformers and transmission

An ideal transformer relates voltages and turns by

V2/V1 = N2/N1.

With negligible loss, V1I1 ≈ V2I2. Raising voltage allows the same power to be sent with smaller current. Since line loss is I²R, high-voltage transmission greatly reduces heating losses. Voltage is then stepped down for distribution and use.

High voltage requires strict insulation, clearance, and protective equipment. Transmission hardware is not a safe setting for amateur experiments.

Household parallel circuits

Household appliances are connected mainly in parallel so each receives the supply voltage and can operate independently. Adding loads increases total current. Circuit breakers interrupt excessive current; residual-current or ground-fault devices detect leakage and reduce shock and fire risk.

Protective earth provides a low-resistance fault path and helps protection operate. Do not remove grounding pins or improvise adapters that defeat safety features.

Electrical and battery safety

Current through the body depends on voltage, skin condition, contact area, path, and time. Wet skin often has lower resistance, increasing danger.

Do not open outlets, distribution panels, or mains-powered equipment. If a device smells unusual, overheats, sparks, changes color, or has a damaged cord, stop using it safely and seek qualified service. Do not throw water on energized electrical equipment.

Lithium-ion batteries store substantial energy. Stop using a battery that is swollen, crushed, leaking, unusually hot, or damaged. Use compatible chargers, avoid high heat, and follow local collection rules. Discarded batteries mixed with ordinary waste can ignite during collection and processing.

Worked example — kettle efficiency

A 1.2 kW kettle runs for 150 s and heats 0.80 kg of water from 20°C to 70°C.

Input energy:

Ein = (1.2 × 10³)(150) = 1.8 × 10⁵ J.

Energy gained by water:

Q = (0.80)(4.2 × 10³)(50) = 1.68 × 10⁵ J.

Efficiency:

η = 1.68/1.8 ≈ 0.93, or 93%.

The remainder mainly heats the kettle and surroundings.

Check your understanding

1. Find the power of a 12 V device drawing 2.0 A.

2. Find the current of a 500 W device at 100 V.

3. Find the energy in kWh used by a 2.0 kW device for 1.5 h.

4. Why does higher transmission voltage allow lower current for the same power?

Answers

1. 24 W.

2. 5.0 A.

3. 3.0 kWh.

4. Because I = P/V when power is fixed.

Chapter 13 — Energy and Society

Energy reaches us through conversion chains

Before a lamp turns on, an energy source, generating equipment, transmission, substations, distribution, and building wiring may all be involved. Primary energy sources include fuels, moving water, wind, sunlight, geothermal heat, and nuclear energy. Electricity and hydrogen are energy carriers produced from primary sources.

Electricity is not a primary source. It is a controllable form that moves energy between places and devices. A generator converts mechanical energy into electrical energy. A photovoltaic cell converts light directly into electrical energy. Appliances then convert electricity into light, heat, motion, or computation.

Every conversion directs some energy into forms other than the intended output. Fair comparison looks beyond operation to extraction, manufacturing, construction, storage, transmission, maintenance, and end-of-life management.

Conserved energy can become less useful

Energy is conserved, so why save it? Although quantity is conserved, highly useful forms can become dispersed low-temperature heat that is difficult to turn back into work.

Brakes convert organized kinetic energy mainly into internal energy. Regenerative braking recovers part of the motion as electrical energy, but resistance, friction, electronics, and storage still produce losses. Efficiency work aims to preserve useful energy quality and reduce unnecessary conversion, not prevent energy from existing.

Thermal generation

Thermal power stations convert chemical or nuclear energy into heat, use a working fluid to turn a turbine, and drive a generator. Combustion-based generation can be dispatchable, but it can also produce carbon dioxide, air pollutants, fuel-supply impacts, and waste heat.

No heat engine converts all input heat into work. Temperature difference, design, and friction limit efficiency. Combined-cycle plants use hot exhaust from a gas turbine to make steam for a second turbine. Combined heat and power uses otherwise wasted heat for buildings or industry.

Hydro, wind, solar, geothermal, and biomass

Hydropower converts gravitational and kinetic energy of water. It can provide flexible output and storage through pumped hydro, but dams affect rivers, sediment, ecosystems, and communities.

Wind turbines convert moving-air energy into rotation. Output varies strongly with wind speed and site. Grid connection, maintenance, sound, landscape, and wildlife interactions must be considered.

Photovoltaic systems convert light directly into electricity without fuel combustion during operation. Output varies with daylight, clouds, season, orientation, and temperature. Manufacturing, land use, recycling, transmission, demand response, and storage are part of the full system.

Geothermal power draws on Earth’s internal heat and is less weather-dependent, but suitable sites are limited and subsurface fluids require careful management.

Biomass uses plant- or waste-derived material. It is not automatically carbon-neutral. Land-use change, transport, processing, regrowth time, and competing uses affect the lifecycle balance.

Nuclear energy and radiation

Nuclear fission releases energy from changes in atomic nuclei. A nuclear plant uses this heat to make steam, turn a turbine, and drive a generator. Operational combustion emissions are low, while accident prevention, spent fuel, radioactive waste, decommissioning, security, and long-term governance remain major responsibilities.

Ionizing radiation includes alpha, beta, gamma, X-ray, and neutron radiation. Their penetration and ionization differ, so protection differs. Alpha radiation is stopped easily outside the body but can be significant if alpha-emitting material enters the body. Gamma radiation penetrates more deeply and requires dense or thick shielding.

Units describe different quantities:

• Bq: nuclear decays per second

• Gy: energy absorbed per kilogram

• Sv: dose adjusted for biological effect.

These numbers are not interchangeable. A half-life is the time for the number of unstable nuclei to fall to half. After four half-lives, one-sixteenth remains; it does not suddenly become zero.

Core protection principles are to reduce time, increase distance, and use appropriate shielding. Medical use balances diagnostic or treatment benefit against risk and should follow qualified professional guidance.

Fusion research

Fusion combines light nuclei and powers the Sun. A terrestrial energy system must confine very hot plasma, sustain the necessary conditions, protect materials, manage fuel cycles, and deliver reliable net electricity at practical cost.

Research progress is real, but experimental plasma performance, whole-facility energy balance, continuous operation, and commercial power are different milestones. Publication schedules and future capacity should not be presented as settled facts without current primary sources.

Balancing an electrical grid

Electricity supply and demand must remain closely matched. Large imbalances affect frequency and voltage stability.

With more variable wind and solar, a grid may combine geographic interconnection, forecasting, flexible generation, demand response, storage, and controlled curtailment. Storage options include batteries, pumped hydro, heat, compressed air, and hydrogen. Each differs in efficiency, response time, capacity, lifetime, materials, location, and cost.

There is rarely one universally best technology. The relevant question is which portfolio fits the time scale, place, reliability requirement, and environmental limits.

Five lenses for comparison

1. Reliability — Can energy be delivered when needed?

2. Environmental impact — What are lifecycle effects on climate, air, water, land, and ecosystems?

3. Safety — How likely are failures, and how large could their consequences be?

4. Economics — What are construction, fuel, operation, grid, storage, and end-of-life costs?

5. Resources and fairness — Who receives benefits, who carries risks, and how durable is the resource base?

A single metric rarely captures the complete decision. Define the boundary, time horizon, and assumptions before comparing numbers.

Calculating a household change

Replace a 60 W lamp with a 9 W lamp providing suitable illumination. If used 5 h per day for 300 days, the annual difference is

(60 - 9) W × 5 h × 300

= 76,500 Wh

= 76.5 kWh.

A full decision also considers brightness, lifetime, purchase cost, manufacturing, and disposal. For heating and cooling, insulation, solar gain, air leakage, ventilation, humidity, and equipment performance all matter.

Check your understanding

1. Convert 2.0 kWh to joules.

2. Name the unit of activity and the unit of absorbed dose.

3. What fraction remains after four half-lives?

4. Why combine transmission, storage, flexible demand, and several generation types?

Answers

1. 7.2 × 10⁶ J.

2. Bq for activity; Gy for absorbed dose.

3. 1/16.

4. To balance changing supply and demand while maintaining grid stability.

Chapter 14 — Using Physics in an Investigation

Turn curiosity into a measurable question

“How does a pendulum move?” is too broad for one experiment. “How does pendulum period change when length changes?” identifies a variable to control and a quantity to measure. A stronger form is: “With bob mass and small release angle held approximately constant, how is period T related to length L?”

Before experimenting, decide:

• what will change

• what will be measured

• what will be kept constant

• what range will be tested.

Changing one main variable at a time makes interpretation clearer.

Hypotheses and predictions

A hypothesis is a provisional explanation. A prediction is a specific outcome expected if that explanation is useful. The hypothesis “a longer pendulum swings more slowly” can lead to the prediction “multiplying length by four will approximately double the period for small angles.”

Results that disagree with a prediction may reveal a flaw in the hypothesis, an uncontrolled variable, a measurement problem, or an invalid approximation. Results that agree support the hypothesis within the tested range; they do not prove it forever.

Record inconvenient results. If a value is excluded, state an objective reason established independently of whether it improves the graph.

Variables

The independent variable is deliberately changed. The dependent variable is measured in response. Controlled variables are kept as constant as practical.

For a cart on an incline, slope angle might be independent and acceleration dependent. The cart, surface, travel distance, release method, and timing system should be controlled.

Not every condition can be held perfectly. Prioritize variables likely to affect the result most strongly, such as friction, sensor alignment, temperature, and battery condition.

Safety belongs in the design

Identify hazards before collecting data: falling masses, moving carts, hot materials, electricity, loud sound, broken glass, and privacy concerns in video or audio.

Follow school instructions, equipment ratings, protective measures, and stop conditions. Do not improvise experiments with household mains, high voltage, large batteries, flame, or radiation sources. A successful investigation is one that can be repeated without exposing people or property to unnecessary risk.

Zero, range, resolution, and calibration

Check the instrument zero before use. Choose a range that safely includes expected values without sacrificing too much resolution. A digital display with many digits is not automatically accurate; sensor precision, calibration, response time, and sampling rate still limit the result.

Compare with a known reference when possible. Document any correction rather than silently adjusting values.

Repeated measurements

For repeated values x1 through xn, the mean is

xmean = (x1 + x2 + … + xn)/n.

Repetition reveals random scatter and often improves an estimate. It does not eliminate systematic error. A camera that is always tilted or a scale with an incorrect zero shifts every result in the same direction. Systematic problems require calibration or redesign.

Graphs, slopes, and linearization

Place the independent variable on the horizontal axis and dependent variable on the vertical. Label quantities and units. Do not simply connect every noisy point; use a best-fit line or curve that represents the trend.

The slope of a line through (x1, y1) and (x2, y2) is

slope = (y2 - y1)/(x2 - x1).

Slope has units and physical meaning. On a force-versus-acceleration graph, slope has units N/(m/s²) = kg and represents mass.

Curved relationships can sometimes be linearized. Free fall gives y ∝ t², so y against t² is linear. A small-angle pendulum gives T ∝ √L, so T² against L is linear. Slope can then estimate a parameter such as g.

Uncertainty and honest precision

Do not report g = 9.812345 m/s² if the experiment distinguishes only tenths of m/s². Match the result’s digits to the measurement.

If repeated values are 9.6, 9.8, and 10.0 m/s², report a mean of 9.8 m/s² together with a spread such as approximately ±0.2 m/s², explaining how the uncertainty was estimated.

Relative error compared with a reference is

relative error = |measured - reference|/|reference|.

Multiply by 100% for percentage error. Remember that a reference value also belongs to stated conditions.

Correlation is not automatically causation

Two quantities can change together because of a third variable. Hot weather may increase both ice-cream sales and swimming incidents; ice cream does not cause the incidents. Causal claims require controlled comparison, repeatability, and a plausible mechanism.

Use language precisely: “associated with” is not the same as “caused by.”

State the model’s limits

Every model omits something. Constant-acceleration equations neglect changing acceleration. Mechanical-energy conservation may neglect friction. An ideal voltmeter draws no current.

A strong report states the range and assumptions: “Air resistance was neglected, so the model is unsuitable for flat paper,” or “Heating may have changed the resistance.” Limits make a conclusion more trustworthy because they define what it actually supports.

Investigation example — analyzing walking motion

Question: How does velocity change from the first step until a walker reaches an approximately steady pace?

Place distance markers along a safe straight path. Fix the camera perpendicular to the motion and choose one body point to track. Read position at known frame times, plot position against time, and calculate interval velocities.

Likely uncertainties include perspective, marker spacing, frame rate, and natural body motion. Repeat several trials. Keep the path clear and do not share identifiable video without permission.

A practical report structure

1. Title and question

2. Hypothesis and reasoning

3. Variables and apparatus

4. Method and safety controls

5. Results, including raw data, units, tables, and graphs

6. Discussion, including patterns, theory, uncertainty, and alternative explanations

7. Conclusion that directly answers the question within limits

8. Sources for data, methods, images, and software.

Separate result from interpretation. “The velocity was 2.1 m/s” is a result. “The value was close to the model because friction was small” is interpretation.

Check your understanding

1. What is the variable deliberately changed?

2. Why repeat measurements?

3. What are the unit and meaning of slope on a position–time graph?

4. Should values that disagree with a hypothesis be deleted?

Answers

1. The independent variable.

2. To measure random scatter and improve an estimate.

3. m/s; velocity.

4. No. Record and investigate them. Exclude only with a documented objective reason.

End Matter and Review

A — Quantities, symbols, and SI units

Quantity Common symbol SI unit Unit symbol position, displacement, length x, y, s, L meter m time, period t, T second s mass m kilogram kg velocity, wave speed v meter per second m/s acceleration a, g meter per second squared m/s² force F, N, W newton N work, heat, energy W, Q, E, K, U joule J power P watt W absolute temperature T kelvin K specific heat capacity c joule per kilogram kelvin J/(kg·K) frequency f hertz Hz wavelength λ meter m charge Q coulomb C current I ampere A voltage V volt V resistance R ohm Ω resistivity ρ ohm meter Ω·m activity A becquerel Bq absorbed dose D gray Gy equivalent or effective dose H, E sievert Sv

Context matters when a symbol has more than one use. W may mean work as a quantity or watt as a unit symbol. N may represent a normal force or the newton unit.

Useful conversions include:

• 1 km = 10³ m

• 1 h = 3,600 s

• 1 km/h = 1/3.6 m/s

• 1 g = 10^-3 kg

• 1 kWh = 3.6 × 10⁶ J

• 1 A = 1 C/s

• 1 V = 1 J/C

• 1 W = 1 J/s

• 1 Ω = 1 V/A.

Area and volume conversion factors must be squared or cubed. Because 1 cm = 10^-2 m, 1 cm² = 10^-4 m² and 1 cm³ = 10^-6 m³.

B — Formula map and conditions

Motion

Average velocity: v = Δx/Δt

Constant velocity: x = x0 + vt

Average acceleration: a = Δv/Δt

Constant acceleration:

v = v0 + at

x = x0 + v0t + 1/2 at²

v² - v0² = 2a(x - x0)

Forces

Weight near Earth: W = mg

Newton’s second law: Fnet = ma

Ideal spring: F = kx

Maximum static friction: fmax = μsN

Kinetic friction: fk = μkN

Work and energy

Work by a constant force: W = Fs cos θ

Power: P = W/t, and for parallel force and velocity, P = Fv

Kinetic energy: K = 1/2 mv²

Gravitational potential energy: U = mgh

Elastic potential energy: U = 1/2 kx²

Work–energy theorem: Wnet = ΔK

Mechanical-energy conservation: K1 + U1 = K2 + U2, when no nonconservative transfer crosses the chosen system boundary.

Efficiency: η = useful output/input

Thermal physics

Absolute temperature: T[K] = t[°C] + 273.15

Temperature change: Q = mcΔT

Phase change: Q = mL

First-law sign convention used here: ΔU = Q + W, with heat into and work done on the system positive.

Waves and sound

Frequency and period: f = 1/T

Wave speed: v = fλ

String and open-pipe modes: fn = nv/(2L)

Ideal closed-pipe fundamental: f1 = v/(4L)

Beat frequency: fbeat = |f1 - f2|

Echo distance: d = vΔt/2

Electricity

Current: I = Q/t

Voltage: V = W/Q

Ohm’s law: V = RI

Series resistance: R = R1 + R2 + …

Parallel resistance: 1/R = 1/R1 + 1/R2 + …

Power: P = VI = I²R = V²/R

Electrical energy: E = Pt = VIt

Ideal transformer: V2/V1 = N2/N1

C — A seven-step solution method

1. Restate the event. Reduce the wording to its physical core: constant speed, smooth incline, parallel circuit, insulated mixing, and so on.

2. Choose the object or system. Decide what is inside the boundary.

3. Choose direction and reference. State the positive direction, zero position, or zero potential energy.

4. Draw. Use a free-body diagram, circuit, waveform, energy chain, or graph.

5. List knowns and unknowns with units. Convert to SI where appropriate.

6. Select a law together with its conditions. Write why constant-acceleration equations, energy conservation, or Ohm’s law applies.

7. Check. Verify units, direction, magnitude, significant figures, and limiting behavior.

D — Thirty common mistakes

1. Dropping units during calculation.

2. Mixing km/h with m/s.

3. Confusing distance with displacement.

4. Confusing speed with velocity.

5. Reading graph height without reading the axes.

6. Treating a negative velocity as “slow.”

7. Treating negative acceleration as automatic slowing.

8. Forgetting initial position x0.

9. Using constant-acceleration equations when acceleration changes substantially.

10. Setting acceleration to zero at the top of a vertical throw.

11. Drawing a force only because an object is moving.

12. Substituting one force into F = ma instead of the net force.

13. Assuming N = mg in every situation.

14. Assuming static friction always equals μsN.

15. Drawing both members of an action–reaction pair on one object.

16. Assuming zero net force means the object must be at rest.

17. Ignoring the angle between force and displacement in work.

18. Treating kinetic energy as proportional to speed instead of speed squared.

19. Saying friction destroys energy.

20. Treating temperature and heat as the same quantity.

21. Using mcΔT during a phase change without including latent heat.

22. Assuming higher source frequency always means higher wave speed.

23. Reading a longitudinal-wave graph as vertical particle motion.

24. Calling adjacent-node distance one wavelength instead of half a wavelength.

25. Forgetting the round trip in an echo problem.

26. Making electron drift direction the same as conventional current in a metal.

27. Saying voltage flows.

28. Adding parallel resistances directly.

29. Claiming larger R always means larger power without stating what is held constant.

30. Removing data solely because they disagree with the expected result.

E — Mixed review problems

1. Out-and-back motion

A cyclist travels 600 m east in 120 s, then 150 m west in 30 s. Find average velocity and average speed for the whole trip.

2. Starting train

A train accelerates from rest at 0.80 m/s² for 15 s. Find final velocity and displacement.

3. Braking

A car at 20 m/s brakes with constant acceleration -5.0 m/s². Find stopping time and braking distance, excluding reaction time.

4. Pulling a box

A 6.0 kg box is pulled 30 N right while friction is 12 N left. Find acceleration.

5. Connected blocks

Blocks of 2.0 kg and 4.0 kg are connected on a frictionless surface. An 18 N force pulls the 2.0 kg block away from the 4.0 kg block through the connecting string. Find acceleration and string tension.

6. Vertical launch

An object is launched upward at 24.5 m/s. Find time and height at the top using g = 9.8 m/s².

7. Work at an angle

A 20 N force at 60° above horizontal moves an object 5.0 m horizontally. Find the work by this force.

8. Energy on a slope

An object starts from rest 10 m above the bottom of a frictionless slope. Find its speed at the bottom.

9. Mechanical-energy loss

A 1.5 kg object descends from 8.0 m and reaches 10 m/s. Find the difference between initial potential energy and final kinetic energy.

10. Heating water

How much energy raises 0.30 kg of water from 15°C to 55°C? Use c = 4.2 × 10³ J/(kg·K).

11. Mixing water

Mix 0.20 kg of water at 70°C with 0.10 kg at 10°C. Ignore the container and surroundings. Find final temperature.

12. Wave quantities

A wave has f = 8.0 Hz and λ = 0.75 m. Find speed and period.

13. Standing wave

Three half-wavelengths fit into a 1.2 m string. Find wavelength.

14. Sonar depth

A sound pulse returns from the seafloor after 1.6 s. Use sound speed 1.5 × 10³ m/s. Find depth.

15. Charge transfer

A current of 0.25 A flows for 2.0 min. Find charge transferred.

16. Series circuit

Resistors of 4.0 Ω and 6.0 Ω are in series across 20 V. Find current and each voltage drop.

17. Parallel circuit

Resistors of 12 Ω and 6.0 Ω are in parallel across 12 V. Find branch currents, total current, and equivalent resistance.

18. Electrical energy

A 1.5 kW heater runs for 40 min. Find energy in kWh and J.

19. Transformer

An ideal transformer has N1 = 1,000, N2 = 100, and V1 = 200 V. Find V2.

20. Investigation design

For an investigation of hanging mass versus spring extension, name an independent variable, dependent variable, controlled variable, and safety control.

F — Review answers

1. Displacement = +450 m, total time = 150 s, so average velocity = +3.0 m/s. Distance = 750 m, so average speed = 5.0 m/s.

2. v = 12 m/s; x = 90 m.

3. 4.0 s; 40 m.

4. Net force 18 N; a = 3.0 m/s² right.

5. Total mass 6.0 kg; a = 3.0 m/s². Tension on the 4.0 kg block is 12 N.

6. 2.5 s; about 31 m.

7. 50 J.

8. 14 m/s.

9. Initial U = 117.6 J; final K = 75 J; difference about 43 J.

10. 5.0 × 10⁴ J.

11. 50°C.

12. 6.0 m/s; 0.125 s.

13. 0.80 m.

14. 1.2 × 10³ m.

15. 30 C.

16. 2.0 A; 8.0 V and 12 V.

17. 1.0 A and 2.0 A; 3.0 A total; 4.0 Ω.

18. 1.0 kWh = 3.6 × 10⁶ J.

19. 20 V.

20. Example: hanging mass independent, extension dependent, same spring controlled; remain below elastic and load limits and keep clear of falling masses.

G — Learning checklist

Can you:

• convert units and use scientific notation?

• distinguish distance from displacement and speed from velocity?

• interpret slope and area on motion graphs?

• choose among constant-acceleration equations by known quantities?

• draw only the forces acting on one selected object?

• distinguish equilibrium from action–reaction pairs?

• apply Fnet = ma with consistent signs?

• explain why acceleration remains nonzero at the top of a throw?

• use work–energy and state when mechanical energy is conserved?

• distinguish temperature, heat, and internal energy?

• separate warming from phase change in thermal calculations?

• connect frequency, period, wavelength, and wave speed?

• identify nodes and antinodes in strings and air columns?

• calculate series and parallel circuits?

• connect voltage, current, power, and energy?

• compare energy technologies with several criteria?

• design an investigation with variables, uncertainty, and safety?

Mark each item “explain independently,” “solve with an example,” or “review needed.” Return only to the topics marked for review.

H — Four-week review route

Week 1: Motion. Read Chapters 1–3. Draw one position–time and one velocity–time graph each day. Complete review problems 1–3.

Week 2: Forces and energy. Read Chapters 4–7. Draw one free-body diagram from daily life each day. Complete problems 4–9 and state the assumptions for every equation.

Week 3: Heat, waves, sound, and circuits. Read Chapters 8–12. Explain unit relationships aloud. Complete problems 10–19.

Week 4: Systems and investigation. Read Chapters 13–14. Estimate one appliance’s daily energy from its rating without opening or modifying it. Design problem 20 as a complete safe investigation.

I — Mathematics bridge

Rearrange symbols before substituting. From V = RI, write R = V/I. From Q = mcΔT, write m = Q/(cΔT). Parentheses protect the full denominator.

State what is constant in proportional reasoning. From K = 1/2 mv², K ∝ v² only when mass is constant. From P = V²/R, P ∝ 1/R only when voltage is constant.

Slope is vertical change divided by horizontal change and has units. Area under a graph has units equal to the product of its axes. On a velocity–time graph, area has units m; on a force–position graph, area has units J.

Use extreme cases to check components. On an incline, mg sin θ should become zero at θ = 0, while mg cos θ should become mg. This test helps catch reversed sine and cosine.

J — Scope and update note

This book was prepared to reflect the current Japanese high school Physics Basics framework as of August 11, 2026, while adapting pedagogy and language for English-reading learners. It is an original work and does not reproduce the prose, exercises, or artwork of a commercial textbook.

Curriculum scope was checked against Japan’s Ministry of Education, Culture, Sports, Science and Technology commentary for the 2018 High School Courses of Study, Science and Mathematics. Digital publishing design was checked against current KDP content, quality, navigation, and reflowable-image guidance.

Regulations, utility tariffs, product specifications, public policy, and technology deployment can change. Check current primary sources before publication or real-world decisions. Safety explanations here are educational and do not replace qualified electrical, medical, radiation-protection, engineering, or emergency guidance.

Closing Note

Foundational physics does not end with a formula sheet. Velocity grows from change in position; acceleration from change in velocity. Net force determines acceleration, and work transfers energy. Heat, waves, sound, and electricity look different, yet all can be read through change, transfer, interaction, and conservation.

When a problem resists you, do not treat that moment as a verdict on your ability. Often one decision is still missing: the positive direction, the system boundary, a unit conversion, or the condition that allows an equation to be used. Draw, label, and translate the equation back into words.

The real world is less tidy than a textbook exercise. Friction and drag exist. Data scatter. Every energy system carries benefits and costs. That is why the habits of physics matter: state assumptions, measure honestly, compare evidence, and define the limits of a conclusion.

Rain on a window, a train entering a station, steam over a cup, sound from a speaker, and the small light on a charger are all readable events. Turn the next “why?” into a question that can be observed or measured. At that point, learning becomes more than preparation for an answer key. It becomes a way to test the world for yourself.

Short Story — One Second Before the Raindrop Stopped

Rain crossed the laboratory windows at an angle, drawing silver lines over the dark schoolyard.

Mio Takase stood by the last bench with a clear storage tub in both hands. Beyond the rooftop, thick clouds overlapped like sheets of charcoal paper. The sky flashed. The thunder arrived several seconds later, low enough to tremble through the glass.

“About six seconds?” Saki Saeki asked. She had already opened a stopwatch on her phone.

“Five point eight. Maybe two kilometers away, roughly.”

“Numbers make it sound much closer.”

“Which is why the window is closing now,” Mr. Kurokawa said from the back of the room.

The students latched the windows, disconnected mains-powered equipment, and stepped away from the sinks and metal frames. Their teacher had revised the afternoon’s plan because of the storm. Only low-voltage battery equipment would remain in use, and that would stop if the lightning came closer.

In three days, the school would hold an open-house program. The physics club’s exhibit was called A World Inside One Second. One group had a pendulum. Another had a resonance tube. Mio and Saki were building a video station that revealed the motion of a falling drop of water.

Their apparatus looked almost too simple to deserve the word apparatus. A nozzle released one drop at a time above a shallow tub. A ruler pattern stood behind the path. A phone recorded at high frame rate. On screen, a drop that vanished in the blink of an eye became a sequence of small transparent spheres suspended at different heights.

Mio loved those frames.

She did not love the graph beside them.

On yesterday’s graph, vertical distance was plotted against time squared. The points should have followed an approximately straight line. Most did. Three points wandered noticeably to the right, and the slope implied a gravitational acceleration of 11.6 m/s².

“Maybe Earth was trying harder yesterday,” Saki said.

“We could put that on the display. ‘At 4:00 p.m., local gravity increased by eighteen percent.’”

“That would definitely attract attention.”

They laughed, but Mio’s smile faded first.

The previous year, she had measured the period of a pendulum for an inquiry project. Three measurements did not match her expectation, so she removed them and submitted the cleaner graph. She had not thought of it as dishonesty. She had thought failed measurements were supposed to disappear.

After her presentation, Mr. Kurokawa had asked one question.

“What rule did you use to exclude those three values?”

She had no answer.

Since then, unexpected points had seemed less like marks on paper and more like eyes waiting to see what she would do.

Saki tapped the table beside the graph. “You want to delete them again?”

“A little.”

“Honest answer.”

“I’m not going to. We find out why they moved.”

Mr. Kurokawa came to their bench and looked not at the graph but at the apparatus.

“If Earth was not unusually ambitious,” he said, “what did you believe stayed constant that may not have stayed constant?”

“Drop size?” Saki said.

“Release height?” Mio offered.

“Write the possibilities. Test them one at a time.”

He moved on without giving them the answer.

Mio divided a notebook page into two columns: What may have changed and How to test it.

Drop size. Nozzle position. Scale angle. Lighting. Recording frame rate. Camera location. Tracking point on the drop. Choice of the first frame.

“The world is not very cooperative about controlled variables,” Saki said.

“It has never read a lab manual.”

They secured the nozzle. They checked the scale with a level. They used the same battery lamp and locked the phone settings. They agreed to track the bottom edge of the drop instead of sometimes choosing its center and sometimes its brightest reflection.

The result barely improved.

Saki replayed the first video. Near the top of the frame, the drop looked close to the printed scale. Near the bottom, it seemed farther from it.

“Is the camera looking downward?” she asked.

Mio walked around the tripod. They had moved one leg to avoid the table support, leaving the lens slightly above the middle of the fall. Worse, the ruler pattern stood several centimeters behind the plane of the falling drop.

“Perspective,” Mio said.

They looked at each other. The troublesome three points no longer looked like enemies. They looked like a message that had finally become legible.

They raised the tripod until the lens was level with the center of the measured path and set it square to the falling plane. They moved the scale into that same plane. Absorbent sheets went under the tub. Tape marks fixed the tripod feet.

They recorded ten new trials.

Mio advanced the video one frame at a time. She entered each position and watched the graph rebuild itself. The points were not perfect. They never would be. But their trend was clear, and the slope now gave 9.7 m/s².

“There it is,” Mio whispered.

Saki leaned over her shoulder. “Earth has returned to normal service.”

Mio placed the old data on the same graph in another color.

“Let’s display both.”

“The bad one too?”

“Especially that one. Without it, visitors won’t know why camera position matters. A perfect graph only says we got lucky or knew what to hide.”

Mr. Kurokawa studied the two data sets.

“Measurement is not a contest to guess the approved number,” he said. “It is an argument about how much confidence a number deserves.”

“Last year I thought it was a contest.”

He paused. “If last year’s mistake changed this year’s method, then it is no longer the same mistake.”

By evening, the storm had strengthened. Water in the rooftop drainpipes made the corridor walls hum. Club members packed away equipment and left in groups after a safety check.

At the laboratory door, Mio remembered the blank title card for their station and turned back.

Lightning whitened the room.

One, two, three.

Thunder arrived after 3.4 seconds. Closer.

The ceiling lights went out.

Emergency lighting came on after a short delay, tinting the room pale green. Mr. Kurokawa called from the corridor.

“Stay where you are. Do not touch the windows or electrical equipment.”

“I’m by the center bench,” Mio answered.

The mains equipment had already been unplugged. The windows were latched. Only their battery lamp remained, making a white circle on the tabletop.

A single soft tap came from the tub.

One drop left in the nozzle had fallen. The battery lamp was beginning to fade and flickered almost invisibly. For one instant, the rhythm of the light and the falling drop aligned. Mio saw a transparent bead hanging in the dark air.

It had not actually stopped. It had velocity. Gravity still accelerated it. The eye had simply received separate bright moments and joined them into an impossible pause.

“The raindrop stopped,” Mio said to the empty bench.

Saki appeared at the doorway beside Mr. Kurokawa. “What stopped?”

“One drop. Only the way it looked.”

“That’s our title.”

“For what?”

“The exhibit.”

“The exhibit is in three days.”

“Exactly. We have three days to print it.”

Mr. Kurokawa aimed a flashlight at the floor. “You may plan your title after we move to the corridor. The safety check is not complete.”

“Yes, sir,” they said together.

The following morning arrived under a startlingly blue sky.

Mio and Saki placed a new title at the center of their panel:

One Second Before the Raindrop Stopped

Below it, they mounted two graphs: the angled-camera data and the corrected-camera data. Every measured point remained visible.

Mio wrote the final caption:

“A drop that appears still is moving between frames. When data disagree with a model, nature may not be wrong. The disagreement may be telling us how we looked.”

Saki read it. “Very serious.”

“It is a science exhibit.”

“We should add, ‘Earth resumed normal operation.’”

“Rejected.”

On open-house day, children and adults crowded around the screen. The slow-motion drops looked like glass marbles arranged in air.

“It’s frozen!” a child said.

“It looks frozen,” Mio replied, advancing one frame. “But it moves between every pair of pictures.”

The gaps between positions grew larger lower in the frame.

“Why are the gaps bigger?”

“Earth pulls the drop, so its downward speed increases. Each second, the velocity changes by about the same amount.”

The child studied the two graphs. “This one is messy.”

“The camera was at an angle, so the scale did not represent distance correctly.”

“Was it a failure?”

For a moment, Mio saw her old pendulum report.

“Yes,” she said, “but it was a useful failure. It told us what to change.”

The child nodded and pressed the release button again.

By late afternoon, the laboratory became quiet. Mio collected visitor cards from the table. One read, The water was round. Another said, I want to measure how far away thunder is. A card written in large uneven letters stopped her.

There is more time between the pictures, even when we cannot see it.

Saki read over her shoulder. “That may be the best line in the entire exhibit.”

“We have been defeated by a fourth grader.”

“A fourth grader with excellent editorial instincts.”

They emptied the tub, wiped the tables, and removed the batteries. Mio reached for the tape marking the tripod feet, then changed her mind.

“Leaving it?” Saki asked.

“For whoever does the experiment next.”

Saki wrote beside the tape: Camera level. Drop and scale in the same plane.

Mio added another line beneath it.

Do not delete the strange point first.

Outside, a bead of water left on the railing fell to the ground. The motion took less than a second. Yet Mio could now see that brief interval holding position, velocity, acceleration, light, sound, and the uncertainty of the person measuring it.

The world did not hide its answers. It moved quickly, vibrated in small ways, and changed form while it waited for a better question.

One week later, Mio sat with a blank course-choice form on her desk. Physics, engineering, computing, environmental science—each field seemed inviting and impossibly large.

Saki turned the chair in front of her and sat backward on it.

“Still empty?”

“I want to do something that involves measurement. That is too vague.”

“The world is vague before you measure it.”

Mr. Kurokawa entered with their recent tests. On Mio’s paper, a question asked for an explanation of a linearized free-fall graph. She had written that the slope represented g/2. She had also written that a nonzero intercept could reveal an error in choosing the starting time or zero position.

Beside the score, her teacher had added: You are turning disagreement into information.

Saki leaned across the aisle. “The local-gravity project continues.”

“Are you going to call it that until graduation?”

“At least.”

Mio laughed and placed her pen on the course form. Instead of forcing a final answer into the first box, she wrote a small action in the margin: Find programs that combine observation and energy systems.

It might not have been the form’s intended use. It was, however, a question she could investigate.

The lunch bell sounded. Pressure waves traveled from the speaker, reflected in the hallway, and arrived again from a distant classroom. The curtain moved near the open transom. Her hand pressed the desk and felt the desk press back.

She had not simply learned more facts. The world had begun sending more signals.

Mio handed in the form and looked at the faint marks the storm had left on the window.

When the next rain came, she would measure again.

No raindrop would ever follow exactly the same path. But once a person learned how to form a question, any brief moment could become a beginning.

Four-Panel Epilogue — Beyond the Falling Drop A four-panel comic in which Mio and Saki investigate inconsistent falling-drop data, correct the camera and scale arrangement, and share the experiment with a child at a school exhibition.

When results disagree with a prediction, do not erase them automatically. Check the measurement method, apparatus, conditions, and recording procedure. A discrepancy is not merely a mark of failure. It may be a clue pointing toward a condition you had not yet noticed.