Quick answer: In calculus-based kinematics, velocity is the derivative of position and acceleration is the derivative of velocity. This AP Physics C lab on calculus kinematics gives the aim, the equipment, the safety points, a step-by-step method, sample results and 7 analysis questions; the answers are on a separate page and the lab sheet is a free printable PDF.
How does the acceleration of a falling object change as it speeds up? You will extract velocity and acceleration from position data of a falling coffee filter stack and relate the changes to air resistance.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:M Unit 1
Aim
Use video measurements of a falling stack of coffee filters to find velocity and acceleration from position data, and show how the acceleration falls as air resistance grows.
Equipment
- four nested basket coffee filters
- vertical metre rule fixed to a wall
- phone with a slow-motion video mode (at least 120 frames per second) on a stand
- stable step stool or tall ladder, used safely
- video player that can step frame by frame
Safety
Do not stand on a chair. Use a stable step stool with a helper, or drop from a high shelf by hand. Keep the floor clear of tripping hazards and wipe up any dust.
Method
- Fix the rule vertically and place the phone 1.5 m away, level with the middle of the rule.
- Release the filters from rest next to the top of the rule while filming.
- Step through the video and, for times 0.00, 0.05, 0.10 … 0.50 s after release, record how far the bottom of the stack has fallen.
- Repeat three times and use the average position at each time.
Results
Sample measurements by Wanjiru in Nairobi:
| t (s) | 0.0 | 0.05 | 0.1 | 0.15 | 0.2 | 0.25 | 0.3 | 0.35 | 0.4 | 0.45 | 0.5 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| distance fallen y (m) | 0.000 | 0.015 | 0.053 | 0.105 | 0.165 | 0.229 | 0.297 | 0.367 | 0.438 | 0.509 | 0.581 |
Analysis questions
Question 1 Average velocities
Calculate the average velocity in each 0.05 s interval and assign it to the midpoint time.Question 2 Velocity–time graph
Plot these velocities against the midpoint times and draw a smooth curve.Question 3 Acceleration at the start and end
Estimate the acceleration for the first two intervals and for the last two intervals.Question 4 Terminal velocity
Estimate the terminal velocity from the graph.Question 5 Check by integration
Add the average velocities × 0.05 s over the whole fall and compare with the measured distance fallen at 0.50 s.Question 6 Explain the shape
Explain why the acceleration decreases with time, using forces.Question 7 Uncertainty
Identify the main source of uncertainty in y and suggest one improvement.
