AP Physics C: Mechanics, AP exams

Unit 1: Kinematics

Unit 1 recasts motion in the language of calculus. Position, velocity, and acceleration are related by differentiation and integration rather than by a fixed set of equations, which lets the course handle non-constant acceleration from the start. Two-dimensional motion and reference frames complete the unit.

Position: 1 of 7 units
Topics covered: 5 topics in this guide

What is covered

Topics inside this unit.

These topics describe the scope students most often review for this part of AP Physics C: Mechanics. Treat them as a study checklist rather than an exhaustive list of what a single exam form can sample.

Describing motion with calculus: Velocity as the derivative of position and acceleration as the derivative of velocity, with the integrals that reverse them.
Kinematics with constant acceleration: Deriving the constant-acceleration relationships rather than memorizing them, including free fall.
Non-constant acceleration: Handling acceleration given as a function of time, position, or velocity, and separating variables when needed.
Motion graphs: Reading slope and area on position, velocity, and acceleration graphs and translating between them.
Two-dimensional and relative motion: Vector decomposition, projectile motion, and describing motion from a different reference frame.

Study tips

How to practice Kinematics.

Section-specific guidance that follows from how this material is usually tested.

Derive the constant-acceleration equations from the integrals once so you can rebuild them if memory fails.
Practice problems where acceleration depends on velocity, since that is where the calculus-based course diverges most from the algebra-based one.
Keep the graph relationships (slope and area) explicit; they appear in both multiple-choice and free-response questions.

Where this fits

Unit 1: Kinematics is 1 of 7 units in the AP Physics C: Mechanics guide. Use the full guide to balance it against everything else you have to review.

Back to the AP Physics C: Mechanics guide Official exam information

Sample Kinematics questions

Practice questions, not yet expert-reviewed

Original practice questions written to the published framework for this section. No expert has reviewed them, and they are not taken from any official exam. Try each one before you open the answer.

Question 1Foundational
A particle moves along the x-axis with position given by x(t) = 3t^2 - 12t + 5, where x is in meters and t is in seconds. What is the particle's velocity at t = 3.0 s?
1. A-3.0 m/s
2. B2.0 m/s
3. C6.0 m/s
4. D18 m/s
Show answer
Answer: C. 6.0 m/s
Velocity is the derivative of position, so v(t) = 6t - 12 and v(3.0) = 18 - 12 = 6.0 m/s. The 18 m/s choice differentiates only the quadratic term and drops the constant -12.
Key insight: Differentiate position for velocity and velocity again for acceleration; no memorized equation is needed.
Question 2Foundational
An object starts from rest at t = 0 and moves along a straight line with acceleration a(t) = 4t, where a is in m/s^2 and t is in seconds. What is its velocity at t = 3.0 s?
1. A6.0 m/s
2. B12 m/s
3. C18 m/s
4. D36 m/s
Show answer
Answer: C. 18 m/s
Integrating acceleration from rest gives v(t) = 2t^2, so v(3.0) = 2(9.0) = 18 m/s. The 12 m/s choice evaluates a(3.0) = 12 m/s^2 and mislabels it a velocity, and 36 m/s integrates 4t as 4t^2 instead of 2t^2.
Key insight: When acceleration is not constant, integrate it rather than reaching for the constant-acceleration equations.
Question 3Intermediate
A particle moves along the x-axis with velocity v(t) = 5 - 2t, where v is in m/s and t is in seconds. What is the particle's displacement between t = 0 and t = 4.0 s?
1. A-4.0 m
2. B4.0 m
3. C8.5 m
4. D20 m
Show answer
Answer: B. 4.0 m
Displacement is the integral of velocity: 5t - t^2 evaluated from 0 to 4.0 gives 20 - 16 = 4.0 m. The 8.5 m choice is the total distance traveled, which counts the backtracking after v changes sign at t = 2.5 s as positive.
Key insight: Displacement is the signed area under a velocity-time graph; distance traveled uses the absolute value.
Question 4Intermediate
A ball is launched from level ground at 20 m/s at an angle of 30 degrees above the horizontal, and air resistance is negligible. Using g = 10 m/s^2, what is the maximum height the ball reaches above the ground?
1. A2.5 m
2. B5.0 m
3. C10 m
4. D20 m
Show answer
Answer: B. 5.0 m
The vertical component is (20 m/s)sin(30 degrees) = 10 m/s, and the rise is v_y^2/(2g) = 100/20 = 5.0 m. The 20 m choice uses the full launch speed instead of its vertical component.
Key insight: Projectile motion is two independent one-dimensional problems that share only the time variable.
Question 5Advanced
A particle moves along the x-axis with position x(t) = t^3 - 6t^2 + 9t, where x is in meters and t is in seconds. At what time after t = 0 does the particle reverse its direction of motion for the second time?
1. A1.0 s
2. B2.0 s
3. C3.0 s
4. D4.0 s
Show answer
Answer: C. 3.0 s
Velocity is v(t) = 3t^2 - 12t + 9 = 3(t - 1)(t - 3), which changes sign at t = 1.0 s and again at t = 3.0 s, so the second reversal is at 3.0 s. The 2.0 s choice is where the acceleration is zero, which is the moment of minimum velocity rather than a reversal.
Key insight: Direction reverses where velocity changes sign, not where acceleration is zero.

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Unit 1: Kinematics