Quick answer: In calculus-based kinematics, velocity is the derivative of position and acceleration is the derivative of velocity. This AP Physics C test on calculus kinematics has 8 questions worth 20 marks, to do in 45 minutes: derivatives of a position function, integrating acceleration and reading a velocity graph. The mark scheme is on a separate page; free printable PDF.
This test covers differentiation and integration of motion, variable acceleration and graph reading. Give every answer with its unit and show the steps of each integral. Marks per part are shown in brackets.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:M Unit 1
Question 1 Derivatives of a position function (3 marks)
A particle has x = t³ − 6t² + 9t (m).- Find v(t) and a(t). (2 marks)
- Find the times at which the particle is at rest. (1 mark)
Question 2 Integrating acceleration (2 marks)
A cart has a = 6t (m/s²) and v(0) = 2 m/s.- Find v(t). (1 mark)
- Calculate v at t = 3 s. (1 mark)
Question 3 Reading a velocity graph (3 marks)
The graph shows the velocity of a trolley.- Calculate the displacement from 0 to 14 s. (2 marks)
- Calculate the acceleration at t = 2 s. (1 mark)
Question 4 Exponential slowing (3 marks)
A puck has a = −0.50v and v(0) = 12 m/s, so v = 12e^(−0.50t).- Calculate v at t = 2.0 s. (1 mark)
- Calculate the total distance travelled before it stops, by integrating v. (2 marks)
Question 5 Acceleration as a function of position (3 marks)
A mass has a = −9x (SI units) and v = 6.0 m/s at x = 0.- Use a = v dv/dx to find the largest value of x. (2 marks)
- Calculate v at x = 1.0 m. (1 mark)
Question 6 Why constant-acceleration equations fail (2 marks)
A student uses x = ½at² for a motion where a = 12t.- Explain why this is wrong. (1 mark)
- State what the area under the a–t graph represents. (1 mark)
Question 7 Speeding up or slowing down (2 marks)
For x = t³ − 6t² + 9t (m), v = 3t² − 12t + 9 and a = 6t − 12.- State whether the speed is increasing or decreasing at t = 2.5 s, using v and a. (1 mark)
- State the same for t = 4.0 s. (1 mark)
Question 8 Simple harmonic motion (2 marks)
A particle has x = 0.30 sin(4.0t) (m).- Calculate the maximum speed. (1 mark)
- Calculate the maximum acceleration. (1 mark)
