AP® Physics 1: Algebra-Based review sheet from Aim for Five (aimforfive.com/physics/units/1/1-2)
Unit 1 · Topic 1.2
1.2 Displacement, Velocity, and Acceleration
Displacement, velocity and acceleration are the three quantities that describe motion. Displacement is your change in position, velocity is how fast your position changes, and acceleration is how fast your velocity changes. The averages compare the start and end of a time interval, and over a very short interval they become instantaneous values.
Key terms
- position
- displacement
- distance vs. displacement
- average velocity
- speed
- acceleration
Treating things as objects
In kinematics you usually use the object model: you ignore an object's size, shape and internal parts and treat it as a single point that has a mass. A car, a runner or a planet can each be a point when only its overall motion matters.
Position, displacement and distance
Position (x) is where the object is, measured from the origin of your coordinate system. Displacement is the change in position: Δx = x_f − x_i, where f means final and i means initial. Δ (delta) always means "final minus initial."
Displacement only cares about where you started and where you ended. Distance is the total length of the path. Run once around a 400 m track and you've covered 400 m of distance, but your displacement is zero because you ended where you began.
Average velocity and average speed
Average velocity is displacement divided by the time interval: v_avg = Δx/Δt. It's a vector with the same sign as the displacement, measured in m/s.
Average speed is total distance divided by time. It's a scalar and is never negative. The two are equal only if the object moves in one direction without turning around.
Acceleration
Average acceleration is the change in velocity divided by the time interval: a_avg = Δv/Δt = (v_f − v_i)/Δt. Its unit is (m/s)/s, written m/s². An acceleration of 3 m/s² means the velocity changes by 3 m/s every second.
An object is accelerating whenever its velocity changes, and velocity can change in size, in direction, or both. A car going around a curve at a steady 20 m/s is accelerating because its direction keeps changing. You'll use that idea in circular motion (2.9).
Acceleration points the way the velocity is changing, which is not always the way the object is moving. A car braking while driving east has a westward acceleration.
Keep the units straight
Use SI units: meters, seconds, m/s and m/s². Speeds given in km/h need converting: divide by 3.6, so 90 km/h = 25 m/s and 72 km/h = 20 m/s. Writing the units at every step catches many errors, because an answer for acceleration that comes out in m/s instead of m/s² means a step went wrong.
From average to instantaneous
Averages only use the start and end of an interval. If you shrink the interval to a tiny fraction of a second, the average velocity gets very close to the instantaneous velocity, the velocity at a single moment. A car's speedometer shows instantaneous speed. Instantaneous acceleration works the same way.
If the velocity is constant, the average and instantaneous velocities are the same at every moment. If it changes, they usually differ, and a single average can hide a lot: a car that averages 15 m/s over a trip may have stopped at lights and briefly reached 25 m/s.
On graphs, instantaneous values come from slopes of tangent lines, which you'll use in 1.3.
| Quantity | Definition | Scalar or vector | SI unit |
|---|---|---|---|
| Displacement | Δx = x_f − x_i | vector | m |
| Distance | total path length | scalar | m |
| Average velocity | Δx/Δt | vector | m/s |
| Average speed | distance ÷ time | scalar | m/s |
| Average acceleration | Δv/Δt | vector | m/s² |
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Average speed versus average velocity (classic trap)
A runner completes exactly one lap of a 400 m track in 80 s, finishing at the starting line. Find her average speed and her average velocity.
Show the solutionHide the solution
- Step 1: Average speed = distance ÷ time = 400 m ÷ 80 s = 5.0 m/s.
- Step 2: She ends where she started, so her displacement is Δx = 0.
- Step 3: Average velocity = Δx/Δt = 0 ÷ 80 s = 0 m/s. The trap is reporting 5.0 m/s for both.
Answer: Average speed 5.0 m/s; average velocity 0 m/s
- Example 2Calculator allowed
Acceleration when an object bounces
A ball moving right at 6.0 m/s hits a wall and bounces straight back, moving left at 4.0 m/s. It touches the wall for 0.020 s. Taking right as positive, find the ball's average acceleration during the contact.
Show the solutionHide the solution
- Step 1: Signed velocities: v_i = +6.0 m/s, v_f = −4.0 m/s.
- Step 2: Δv = v_f − v_i = (−4.0) − (+6.0) = −10.0 m/s. The change is 10 m/s, not 2 m/s, because the ball reversed.
- Step 3: a_avg = Δv/Δt = (−10.0 m/s) ÷ (0.020 s) = −500 m/s².
- Step 4: The negative sign means the acceleration points left, away from the wall, which makes sense: the wall pushed the ball back.
Answer: 500 m/s² to the left (−500 m/s²)
- Example 3Calculator allowed
Average velocity from two positions
A cart is at x = +2.0 m at t = 1.0 s and at x = −6.0 m at t = 5.0 s. Find its average velocity over this interval.
Show the solutionHide the solution
- Step 1: Δx = x_f − x_i = (−6.0) − (+2.0) = −8.0 m.
- Step 2: Δt = 5.0 s − 1.0 s = 4.0 s.
- Step 3: v_avg = Δx/Δt = −8.0 m ÷ 4.0 s = −2.0 m/s, so it moved in the negative direction on average.
Answer: −2.0 m/s (2.0 m/s in the negative direction)
Common mistakes
- Using distance instead of displacement for average velocity. Velocity needs Δx = x_f − x_i, which can be zero or negative.
- Computing Δv as the difference in speeds when the object reverses. Subtract signed velocities: a bounce from +6 to −4 m/s is Δv = −10 m/s.
- Thinking constant speed means zero acceleration. A change in direction is also an acceleration.
- Assuming acceleration points in the direction of motion. It points the way velocity is changing, so a braking car accelerates backward.
On the exam
- Expect questions that give positions or velocities at two times and ask for an average, or that compare average speed and average velocity for a round trip.
- When a question says "at the instant" or "at t = 3 s," it wants an instantaneous value, which usually means reading a graph's slope or using a kinematic equation, not dividing totals.
Connected topics
Videos
Check yourself
4 questions on 1.2 Displacement, Velocity, and Acceleration. Pick an answer to see if you got it, and why.
A cart's velocity changes from +12 m/s to −8.0 m/s in 4.0 s. What is the cart's average acceleration?
An elevator is moving downward and slowing to a stop. Upward is positive. Which describes the elevator's velocity and acceleration?
A runner completes exactly one lap of a 400 m track in 80 s, finishing where she started. What are her average speed and the magnitude of her average velocity?
A cyclist rides 600 m at a constant 10 m/s and then another 600 m in the same direction at a constant 30 m/s. What is the cyclist's average speed for the whole trip?
0 of 4 answered