Velocity and acceleration

A lift can move upward while slowing down. “Up” tells us about velocity; “slowing” asks how that velocity changes. Separating those two questions makes motion graphs much less mysterious.

We will describe motion before discussing what causes it. Begin with an origin, a direction and a clock. From a record of positions, build average velocity, then instantaneous velocity, then acceleration. Graphs will help you see why a zero value and a zero rate of change are different statements.

What you will learn to do

Distinguish position, displacement, distance, velocity and speed.
Interpret instantaneous velocity as a local position-time slope.
Determine whether speed is increasing from the directions of velocity and acceleration.
Read displacement from signed velocity-time area and handle reversals explicitly.

Useful before this chapterVectors and their components →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Velocity and acceleration
Learning goals:
- Distinguish position, displacement, distance, velocity and speed.
- Interpret instantaneous velocity as a local position-time slope.
- Determine whether speed is increasing from the directions of velocity and acceleration.
- Read displacement from signed velocity-time area and handle reversals explicitly.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

BEFORE YOU READ

Bring back one idea

Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.

You walk away from a door and return to it. Can the average velocity be zero even though you moved?

Compare your reasoning

Yes. Net displacement is zero, so displacement divided by total time is zero. Distance travelled is positive, so average speed need not vanish.

Revisit the explanation →

01 / Velocity and acceleration

Give the motion a frame and a clock

Position is measured relative to an origin and chosen axes. On a straight track, choose a positive direction and call the coordinate x. A particle at x = −3 m is three metres on the negative side of the origin; the sign tells you where it is, not which way it is moving.

Time has an origin too. You may define t = 0 when a cart passes a gate, even though it was moving before that event. An equation x(t) specifies a position for each clock reading in its stated domain. It does not automatically describe the motion before or after that domain.

Displacement is final position minus initial position. Distance travelled adds the lengths of all parts of the path. A reversal changes how distance accumulates but does not change the endpoint definition of displacement.

\Delta x=x_f-x_i

Always subtract initial from final using the same origin and axis. A negative displacement indicates net change toward the negative direction.

WORKED EXAMPLE

A return journey does not erase distance

A walker moves 8 m right in 4 s, then 3 m left in 2 s. Take right as positive. Find displacement, distance, average velocity and average speed for the complete journey.

Choose an approach

First calculate the signed change in position and the positive path length separately.

The final position relative to the start is +8 − 3 = +5 m. The path length is 8 + 3 = 11 m.
The total elapsed time is 4 + 2 = 6 s.
$v_{\rm avg}=\frac{5}{6}\ \mathrm{m/s}$
Average speed uses the total path length, not the displacement.
$\text{average speed}=\frac{11}{6}\ \mathrm{m/s}$

What this gives usThe average velocity is +5/6 m/s; the average speed is 11/6 m/s.

Check the result another way

The magnitude of average velocity is no greater than average speed. They would match for this straight track if there were no reversal.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: A return journey does not erase distance
Problem: A walker moves 8 m right in 4 s, then 3 m left in 2 s. Take right as positive. Find displacement, distance, average velocity and average speed for the complete journey.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Velocity and acceleration
Focus: Give the motion a frame and a clock
Learning goals:
- Distinguish position, displacement, distance, velocity and speed.
- Interpret instantaneous velocity as a local position-time slope.
- Determine whether speed is increasing from the directions of velocity and acceleration.
- Read displacement from signed velocity-time area and handle reversals explicitly.
Equations in this section (LaTeX):
\Delta x=x_f-x_i
Meaning: Always subtract initial from final using the same origin and axis. A negative displacement indicates net change toward the negative direction.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

02 / Velocity and acceleration

An average belongs to a specified interval

Average velocity compresses an entire interval into one ratio. Different motions can have the same displacement and elapsed time: one particle may move steadily, while another waits and then rushes. The common average does not reveal the complete history.

On a position-time graph, average velocity is the slope of the line connecting the two endpoint events. The vertical change is displacement; the horizontal change is elapsed time. Do not confuse the height of a point with the slope through it.

The arithmetic mean of an initial and final velocity is not always the time-averaged velocity. That shortcut is valid when velocity changes linearly with time, as under constant acceleration. For a general motion, calculate displacement divided by elapsed time or integrate the velocity record.

v_{\rm avg}=\frac{x(t_2)-x(t_1)}{t_2-t_1}

The quotient has units of metres per second. It is a secant slope across the chosen interval.

STOP & THINK

On a position-time graph, a particle is at a positive position but the curve slopes downward. Its velocity is…

PositiveNegativeNecessarily zero

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Velocity and acceleration / An average belongs to a specified interval
Problem: On a position-time graph, a particle is at a positive position but the curve slopes downward. Its velocity is…
Choices:
1. Positive
2. Negative
3. Necessarily zero

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Velocity and acceleration
Focus: An average belongs to a specified interval
Learning goals:
- Distinguish position, displacement, distance, velocity and speed.
- Interpret instantaneous velocity as a local position-time slope.
- Determine whether speed is increasing from the directions of velocity and acceleration.
- Read displacement from signed velocity-time area and handle reversals explicitly.
Equations in this section (LaTeX):
v_{\rm avg}=\frac{x(t_2)-x(t_1)}{t_2-t_1}
Meaning: The quotient has units of metres per second. It is a secant slope across the chosen interval.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

03 / Velocity and acceleration

Shrink the interval to describe one instant

A speedometer refers to what is happening now, not the average over the last hour. To make the idea precise, consider average velocities over shorter and shorter intervals surrounding an instant. If those ratios approach a limit, that limit is the instantaneous velocity.

The derivative notation dx/dt names this limit. It does not mean that both numerator and denominator are simply set equal to zero and then divided. It describes the limiting ratio of their changes. On a smooth position-time graph it is the tangent slope.

v(t)=\lim_{h\to0}\frac{x(t+h)-x(t)}{h}=\frac{dx}{dt}

A tangent describes the local direction of the curve. A horizontal tangent gives zero instantaneous velocity even if motion occurs just before and after.

WORKED EXAMPLE

Watch an average approach an instantaneous value

A particle has x(t) = 2t², with x in metres and t in seconds. Find its instantaneous velocity at t = 1 s without assuming it moves uniformly.

Choose an approach

Calculate the average velocity from 1 to 1 + h, simplify, then let the interval shrink.

Substitute the two positions before dividing.
$\frac{x(1+h)-x(1)}{h}=\frac{2(1+h)^2-2}{h}$
Expand the numerator: 2 + 4h + 2h² − 2. Dividing by nonzero h gives 4 + 2h in the stated SI convention.
For h = 0.1 s the average is 4.2 m/s; for 0.01 s it is 4.02 m/s. The limit is 4 m/s.

What this gives usThe instantaneous velocity at 1 s is +4 m/s.

Check the result another way

Differentiating x = 2t² gives v = 4t, so substitution at 1 s gives the same answer. The finite-interval average need not equal the instantaneous value.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Watch an average approach an instantaneous value
Problem: A particle has x(t) = 2t², with x in metres and t in seconds. Find its instantaneous velocity at t = 1 s without assuming it moves uniformly.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

A physically meaningful model must state where it is smooth enough for the derivative. At an idealized abrupt change in velocity, a single finite instantaneous acceleration may not be defined. The impulse model is useful for such short interactions; pretending every drawn corner is smooth can obscure the approximation.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Velocity and acceleration
Focus: Shrink the interval to describe one instant
Learning goals:
- Distinguish position, displacement, distance, velocity and speed.
- Interpret instantaneous velocity as a local position-time slope.
- Determine whether speed is increasing from the directions of velocity and acceleration.
- Read displacement from signed velocity-time area and handle reversals explicitly.
Equations in this section (LaTeX):
v(t)=\lim_{h\to0}\frac{x(t+h)-x(t)}{h}=\frac{dx}{dt}
Meaning: A tangent describes the local direction of the curve. A horizontal tangent gives zero instantaneous velocity even if motion occurs just before and after.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

04 / Velocity and acceleration

Acceleration tells you how the velocity changes

Acceleration is the rate of change of velocity. In one dimension, its sign tells you the direction in which the velocity is changing. A positive acceleration adds positive velocity each second. If the particle is initially moving left, that can reduce its speed rather than increase it.

The units m/s² mean a change in velocity, measured in m/s, per second. With constant a = +2 m/s², the signed velocity increases by 2 m/s in each second. This statement does not specify the initial velocity, so it does not alone tell you the direction of motion.

For nonzero velocity in one dimension, speed increases when velocity and acceleration have the same sign and decreases when they have opposite signs. At v = 0, inspect the motion immediately before and after rather than trying to attach a direction to a zero vector.

a_{\rm avg}=\frac{\Delta v}{\Delta t},\qquad a(t)=\frac{dv}{dt}

These are average and instantaneous acceleration respectively. On a velocity-time graph they are secant and tangent slopes.

Keep direction and speeding up separate
Velocity	Acceleration	Immediate speed behaviour
+	+	Increasing
+	−	Decreasing
−	+	Decreasing
−	−	Increasing

WORKED EXAMPLE

Negative velocity does not mean negative acceleration

A cart’s velocity changes from −7 m/s to −3 m/s in 2 s. Find its average acceleration and describe the speed change.

Choose an approach

Subtract signed velocities, then compare their magnitudes.

The velocity change is −3 − (−7) = +4 m/s.
Divide by the positive elapsed time.
$a_{\rm avg}=\frac{+4}{2}=+2\ \mathrm{m/s^2}$
The speed decreases from 7 to 3 m/s. The cart still moves left at both endpoints; the positive acceleration opposes that motion.

What this gives usThe average acceleration is +2 m/s², while the speed decreases.

Check the result another way

A sign-only rule such as “negative means slowing” would fail here. Compare velocity and acceleration directions.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Negative velocity does not mean negative acceleration
Problem: A cart’s velocity changes from −7 m/s to −3 m/s in 2 s. Find its average acceleration and describe the speed change.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

STOP & THINK

At the top of a vertical throw with negligible air drag, velocity is zero. The acceleration is…

Zero, because the ball stopsDownward with magnitude gUpward because the ball was rising

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Velocity and acceleration / Acceleration tells you how the velocity changes
Problem: At the top of a vertical throw with negligible air drag, velocity is zero. The acceleration is…
Choices:
1. Zero, because the ball stops
2. Downward with magnitude g
3. Upward because the ball was rising

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Velocity and acceleration
Focus: Acceleration tells you how the velocity changes
Learning goals:
- Distinguish position, displacement, distance, velocity and speed.
- Interpret instantaneous velocity as a local position-time slope.
- Determine whether speed is increasing from the directions of velocity and acceleration.
- Read displacement from signed velocity-time area and handle reversals explicitly.
Equations in this section (LaTeX):
a_{\rm avg}=\frac{\Delta v}{\Delta t},\qquad a(t)=\frac{dv}{dt}
Meaning: These are average and instantaneous acceleration respectively. On a velocity-time graph they are secant and tangent slopes.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

05 / Velocity and acceleration

Connect graph slope, graph area and the actual motion

A position-time graph answers “where?” Its slope answers “how fast and which way?” A velocity-time graph answers that second question directly; its slope gives acceleration. Label both axes before interpreting a line or a shaded region.

For a short interval with nearly constant velocity, displacement is approximately velocity multiplied by time width. On a velocity-time plot this is a thin rectangular area. Adding such contributions and letting the widths shrink gives a signed area, or integral, for total displacement.

Area above the time axis contributes positive displacement; area below contributes negative displacement. Distance instead adds the magnitudes of all those small path contributions. When velocity reverses, cancellation in displacement must not also cancel distance.

\Delta x=\int_{t_1}^{t_2}v(t)\,dt,\qquad d=\int_{t_1}^{t_2}|v(t)|\,dt

For straight-line motion, signed area gives displacement and absolute area gives distance. Both have units (m/s) × s = m.

TRY THE IDEA

Pass through a turning point slowly

Use acceleration −2 m/s² and move the time slider to 3 s, then 6 s. Compare velocity, displacement and distance. Change the acceleration only after explaining the first trajectory.

Time since start: 0.0 sConstant acceleration: -2 m/s²

Position from x₀ = 00.00 m

Velocity6.00 m/s

Total distance0.00 m

The shaded signed area gives displacement. With a = −2 m/s², stop at t = 3 s, then continue to 6 s: the particle reverses, so distance and displacement separate. This is manual exploration; there is no playback clock to keep up with.

WORKED EXAMPLE

Two triangles tell different stories

A particle has v(t) = 6 − 2t m/s for 0 ≤ t ≤ 6 s. Find its displacement and distance travelled.

Choose an approach

Locate the zero of velocity so the positive and negative parts are separated.

The velocity is zero at t = 3 s. The first triangle has height 6 m/s and width 3 s.
Its signed area is +9 m. From 3 to 6 s, the equal triangle lies below the axis and contributes −9 m.
$\Delta x=9-9=0\ \mathrm m$
Distance adds the two positive path lengths.
$d=9+9=18\ \mathrm m$

What this gives usThe particle returns to its starting position after travelling 18 m.

Check the result another way

The final velocity is −6 m/s, so returning to the starting position does not mean returning to rest.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Two triangles tell different stories
Problem: A particle has v(t) = 6 − 2t m/s for 0 ≤ t ≤ 6 s. Find its displacement and distance travelled.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Velocity and acceleration
Focus: Connect graph slope, graph area and the actual motion
Learning goals:
- Distinguish position, displacement, distance, velocity and speed.
- Interpret instantaneous velocity as a local position-time slope.
- Determine whether speed is increasing from the directions of velocity and acceleration.
- Read displacement from signed velocity-time area and handle reversals explicitly.
Equations in this section (LaTeX):
\Delta x=\int_{t_1}^{t_2}v(t)\,dt,\qquad d=\int_{t_1}^{t_2}|v(t)|\,dt
Meaning: For straight-line motion, signed area gives displacement and absolute area gives distance. Both have units (m/s) × s = m.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

06 / Velocity and acceleration

Translate between descriptions before choosing an equation

A motion problem may begin with words, a table, a graph or an equation. Convert enough information to see what is actually specified. “Moving uniformly” means constant velocity only if the direction is fixed as well as the speed. “Momentarily at rest” specifies a velocity at one instant; it does not specify the acceleration.

For a data table, finite differences estimate averages over intervals. They do not determine an arbitrary motion exactly between readings without a model. For a graph, use its scale: a line that looks steep because the axes are stretched is not necessarily a large physical velocity.

Finish with three checks. Have you used the same axes throughout? Do the units match the requested quantity? Does your sign match the direction shown by the motion? The next chapter derives constant-acceleration equations from these slope and area ideas, so they no longer need to be an unrelated list to memorize.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Velocity and acceleration
Focus: Translate between descriptions before choosing an equation
Learning goals:
- Distinguish position, displacement, distance, velocity and speed.
- Interpret instantaneous velocity as a local position-time slope.
- Determine whether speed is increasing from the directions of velocity and acceleration.
- Read displacement from signed velocity-time area and handle reversals explicitly.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

BRING THE IDEAS TOGETHER

What should stay with you

Position is a coordinate; velocity is its rate of change; acceleration is the rate of change of velocity.
Average velocity uses displacement, while average speed uses total path length.
The slope of x(t) is velocity and the slope of v(t) is acceleration.
For nonzero one-dimensional velocity, acceleration in the opposite direction reduces speed.
Integrate signed velocity for displacement and its magnitude for distance.

RETRIEVE, THEN EXPLAIN

Try it without the examples

Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.

SELF CHECK 1

A particle moves from x = −2 m to x = +7 m in 3 s. What is its average velocity?

Your answer (m/s)Use a decimal, a fraction such as 1/3, or scientific notation.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Velocity and acceleration
Problem: A particle moves from x = −2 m to x = +7 m in 3 s. What is its average velocity?
Requested answer unit: m/s

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

SELF CHECK 2

A particle has v < 0 and a < 0 at an instant. Its speed is…

IncreasingDecreasingNecessarily zero

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Velocity and acceleration
Problem: A particle has v < 0 and a < 0 at an instant. Its speed is…
Choices:
1. Increasing
2. Decreasing
3. Necessarily zero

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

SELF CHECK 3

For v(t) = 6 − 2t m/s, what is the speed at t = 4 s?

Your answer (m/s)Use a decimal, a fraction such as 1/3, or scientific notation.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Velocity and acceleration
Problem: For v(t) = 6 − 2t m/s, what is the speed at t = 4 s?
Requested answer unit: m/s

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

SELF CHECK 4

A horizontal line on a position-time graph means…

Constant nonzero speedRest at a fixed coordinateConstant nonzero acceleration

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Velocity and acceleration
Problem: A horizontal line on a position-time graph means…
Choices:
1. Constant nonzero speed
2. Rest at a fixed coordinate
3. Constant nonzero acceleration

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

APPLY THE CHAPTER

Take the ideas into a problem

Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.

What happens at the highest point?Beginner · Free sample · Hints and guided reasoning Can velocity be zero while acceleration is not?Beginner · Hints and guided reasoning Zero velocity with a nonzero slope of velocityIntermediate · Hints and guided reasoning Acceleration from a speed-position lawAdvanced · Hints and guided reasoning A growing accelerationIntermediate · Hints and guided reasoning

COME BACK TO THE IDEA

What can you explain later?

Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.

Next study session

Explain how a body moving left can have acceleration to the right.

Compare your reasoning

Choose right as positive. Its velocity is negative but increasing toward zero, so acceleration is positive and it slows while still moving left. Direction of motion and direction of change are different.

Revisit the explanation →

A few days later: change the situation

At a smooth maximum of a position–time graph, what is zero? Does that settle the acceleration?

Compare your reasoning

The slope, hence instantaneous velocity, is zero. Acceleration depends on how the slope changes. At an ordinary rounded maximum it is negative, but zero slope alone does not determine its value.

Revisit the explanation →

Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.

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