A driver notices a stopped vehicle ahead. First the car continues moving during the reaction interval; then braking changes its speed. Treating the whole event as one formula hides the part where the brakes have not acted yet.
The skill is not remembering which equation contains the right letters. It is deciding what the interval is, what the signs mean, and whether acceleration really stays constant throughout it. We will build the equations from a velocity–time graph, then use them in situations where an unexamined substitution gives the wrong story.
What you will learn to do
- Derive the constant-acceleration equations from a graph.
- Choose signs and starting conditions before substituting numbers.
- Separate a journey into intervals when its acceleration changes.
- Interpret stopping times, reversals and multiple mathematical roots.
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: Motion with constant acceleration Learning goals: - Derive the constant-acceleration equations from a graph. - Choose signs and starting conditions before substituting numbers. - Separate a journey into intervals when its acceleration changes. - Interpret stopping times, reversals and multiple mathematical roots. 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.
What does constant acceleration say about equal time intervals?
Compare your reasoning
Velocity changes by the same signed amount in each equal interval. Position usually does not. That distinction is why a position–time graph is curved even when acceleration is constant.
Revisit the explanation →01 / Motion with constant acceleration
Tell the story before choosing an equation
Choose a straight coordinate axis and mark its positive direction. Let t = 0 be the beginning of the particular interval you are studying. Call the velocity then u, the velocity at elapsed time t v, and the displacement during that interval s. The symbol s is a signed change in position, not automatically a distance travelled.
Constant acceleration means that velocity changes by equal amounts in equal times. It does not mean constant speed. It also does not mean the acceleration has to point in the direction of motion. A negative acceleration is a direction statement; a particle travelling negatively can speed up under it.
| Quantity | Meaning | SI unit |
|---|---|---|
| u | Velocity at the chosen start | m/s |
| v | Velocity at the chosen end | m/s |
| a | Constant acceleration throughout | m/s² |
| t | Elapsed time, with t ≥ 0 | s |
| s | Final position minus initial position | m |
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: Motion with constant acceleration Focus: Tell the story before choosing an equation Learning goals: - Derive the constant-acceleration equations from a graph. - Choose signs and starting conditions before substituting numbers. - Separate a journey into intervals when its acceleration changes. - Interpret stopping times, reversals and multiple mathematical roots. 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.
02 / Motion with constant acceleration
Build the equations from a graph
On a velocity–time graph, constant acceleration produces a straight line. Its slope is the change in velocity divided by elapsed time. Rearranging that definition gives the first equation. Nothing about distance has entered yet.
For t > 0, the slope is a. The rearranged equation also gives v = u at t = 0.
Displacement is the signed area between the velocity graph and the time axis. For a straight segment, that area is the initial-velocity rectangle plus the triangle representing the change in velocity. If parts of the graph lie below the axis, their areas carry negative signs.
The arithmetic mean of the endpoint velocities is the time-average velocity here because the graph is straight. It is not a general averaging rule for arbitrary motion.
Sometimes time is neither known nor needed. Multiply the average-velocity expression for displacement by 2a, and use at = v − u. The two factors become a difference of squares.
This relation loses the sign of v when velocity is squared. Recover that sign from the physical stage of the motion.
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: Motion with constant acceleration
Focus: Build the equations from a graph
Learning goals:
- Derive the constant-acceleration equations from a graph.
- Choose signs and starting conditions before substituting numbers.
- Separate a journey into intervals when its acceleration changes.
- Interpret stopping times, reversals and multiple mathematical roots.
Equations in this section (LaTeX):
a=\frac{v-u}{t}\quad\Longrightarrow\quad v=u+at
Meaning: For t > 0, the slope is a. The rearranged equation also gives v = u at t = 0.
s=ut+\frac12(at)t=ut+\frac12at^2=\frac{u+v}{2}t
Meaning: The arithmetic mean of the endpoint velocities is the time-average velocity here because the graph is straight. It is not a general averaging rule for arbitrary motion.
2as=(u+v)at=(u+v)(v-u)\quad\Longrightarrow\quad v^2=u^2+2as
Meaning: This relation loses the sign of v when velocity is squared. Recover that sign from the physical stage of the motion.
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.03 / Motion with constant acceleration
Solve one interval and check it a second way
WORKED EXAMPLE
A cart accelerates along a straight track
At the start of a 4.0 s interval, a cart moves right at 3.0 m/s. Its acceleration is a constant 2.0 m/s² rightward. Find its final velocity and displacement. Treat the given values as ideal model inputs.
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STOP & THINK
A particle has initial velocity 2 m/s and final velocity 8 m/s. Is its average velocity necessarily 5 m/s?
- Yes, endpoint velocities always average this way
- Only with additional information, such as constant acceleration
- No, average velocity can never be 5 m/s
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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: Motion with constant acceleration Focus: Solve one interval and check it a second way Learning goals: - Derive the constant-acceleration equations from a graph. - Choose signs and starting conditions before substituting numbers. - Separate a journey into intervals when its acceleration changes. - Interpret stopping times, reversals and multiple mathematical roots. 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.
04 / Motion with constant acceleration
A stopping time is a boundary, not a new law
WORKED EXAMPLE
Braking to rest
A vehicle moves at 12 m/s and brakes with a constant acceleration of −3 m/s² until it stops. It then remains at rest. Find its braking time and braking distance.
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A freely moving particle under a sustained force can reverse direction after v becomes zero. A vehicle described as braking to rest and staying there cannot. In both cases the same equation may describe the earlier interval. What happens at its boundary comes from the physical conditions, not from the equation alone.
INTERACTIVE LAB
Follow a genuine reversal
This model keeps its chosen acceleration after velocity reaches zero. Set a = −2 m/s². Predict the position at 3 s and at 6 s, then compare displacement with total distance. Explain why this differs from the vehicle that stays stopped.
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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: Motion with constant acceleration Focus: A stopping time is a boundary, not a new law Learning goals: - Derive the constant-acceleration equations from a graph. - Choose signs and starting conditions before substituting numbers. - Separate a journey into intervals when its acceleration changes. - Interpret stopping times, reversals and multiple mathematical roots. 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.
05 / Motion with constant acceleration
Gravity does not switch off at the top
For motion close to Earth’s surface, neglecting air resistance and taking g as constant, vertical acceleration is downward throughout the flight. Choose upward positive: then a = −g during ascent, at the highest point and during descent. Velocity changes sign; acceleration does not.
At the highest point, vertical velocity is zero at one instant. That does not provide a period of rest. A nonzero downward acceleration immediately gives the ball a downward velocity. If air resistance or large changes of altitude matter, this constant-g model needs revision.
WORKED EXAMPLE
Up and back to the launch height
A ball is thrown vertically upward at 18 m/s. Use g = 10 m/s² and neglect air resistance. Find the greatest rise and the time to return to its launch height.
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STOP & THINK
At the highest point of this ball’s flight, which pair is correct with upward positive?
- v = 0, a = 0
- v = 0, a = −10 m/s²
- v = −10 m/s, a = 0
MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →
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: Motion with constant acceleration Focus: Gravity does not switch off at the top Learning goals: - Derive the constant-acceleration equations from a graph. - Choose signs and starting conditions before substituting numbers. - Separate a journey into intervals when its acceleration changes. - Interpret stopping times, reversals and multiple mathematical roots. 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.
06 / Motion with constant acceleration
Split a journey where the conditions change
WORKED EXAMPLE
Reaction distance plus braking distance
A car moves at 20 m/s. The driver takes 0.60 s to react, maintaining that speed, then brakes at a constant 5.0 m/s² until rest. Find the total stopping distance and time.
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At fixed braking deceleration, the braking distance grows with the square of initial speed, whereas reaction distance grows directly with speed. Doubling speed therefore doubles the reaction part and quadruples the braking part. This comparison assumes the same reaction time and braking conditions; it is not a promise about every road or vehicle.
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: Motion with constant acceleration Focus: Split a journey where the conditions change Learning goals: - Derive the constant-acceleration equations from a graph. - Choose signs and starting conditions before substituting numbers. - Separate a journey into intervals when its acceleration changes. - Interpret stopping times, reversals and multiple mathematical roots. 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.
07 / Motion with constant acceleration
Read a mathematical answer as a physical event
Before calculating, list the known quantities with signs and units. Identify the event that ends the interval: an elapsed time, arrival at a position, a collision, or reaching v = 0. Then select an equation that relates that event to the unknown.
A quadratic position equation can have two valid positive times because a projectile passes a height once going up and once going down. A negative root can describe an extrapolated event before your chosen start. Neither discard nor accept a root solely because it looks inconvenient: check it against the stated time domain and motion.
After calculating, substitute into a different relation or inspect a graph. Check units, signs and limiting cases. If acceleration approaches zero, displacement should approach ut. If your “braking distance” is negative while you chose the initial motion as positive, revisit your signs or the event you solved for.
STOP & THINK
A vehicle travels at 15 m/s for a reaction time of 0.80 s before braking begins. How far does it travel during reaction alone?
Answer in m.
MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →
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: Motion with constant acceleration Focus: Read a mathematical answer as a physical event Learning goals: - Derive the constant-acceleration equations from a graph. - Choose signs and starting conditions before substituting numbers. - Separate a journey into intervals when its acceleration changes. - Interpret stopping times, reversals and multiple mathematical roots. 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.
BRING THE IDEAS TOGETHER
What should stay with you
- Constant-acceleration equations describe one interval with a single constant acceleration.
- Displacement is signed area; total distance requires splitting at reversals.
- A zero velocity does not imply zero acceleration.
- Check whether roots and extrapolated times belong to the physical interval.
- Use a second equation or a graph to test your result.
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 cart starts from rest with a constant acceleration of 3 m/s². How far does it move in 4 s?
Answer in m.
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SELF CHECK 2
A ball rises and returns to its launch height without drag. Which statement about the entire flight is correct?
- Average velocity is zero; average speed is positive
- Both averages are zero
- Average speed equals the launch speed
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SELF CHECK 3
A particle initially moving at −4 m/s has constant acceleration +2 m/s². After how many seconds does it first have zero velocity?
Answer in s.
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SELF CHECK 4
When can v² = u² + 2as give the correct speed but leave a direction undecided?
- Never; a square always fixes the sign
- When both positive and negative velocities are possible at the specified position
- Only when the acceleration is zero
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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.
A puck slows on a rough tableIntermediate · Free sample · Hints and guided reasoningRead the end of the velocity lineBeginner · Free sample · Hints and guided reasoningCan velocity be zero while acceleration is not?Beginner · Hints and guided reasoningHow far does the cart travel before stopping?Intermediate · Hints and guided reasoningInfer the braking accelerationIntermediate · Hints and guided reasoningDistance after a direction changeAdvanced · Hints and guided reasoningJoin two motion stagesIntermediate · Hints and guided reasoningCOME 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
From rest with constant acceleration, why does doubling the elapsed time quadruple displacement?
Compare your reasoning
Average velocity doubles and the duration doubles. Their product therefore grows by four. Equivalently, the area under the straight velocity–time line is a triangle proportional to time squared.
Revisit the explanation →A few days later: change the situation
If initial speed doubles, how do reaction distance and braking distance change for fixed reaction time and fixed braking deceleration?
Compare your reasoning
Reaction distance doubles because speed stays constant during that interval. Braking distance quadruples because it depends on the square of the initial speed. Keep the two stages separate.
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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