Question pattern: Kinematics data analysis
Original question 2 of 2 in this pattern
Estimate gravitational acceleration from video data
A ball is released from rest and filmed as it falls. Downward displacement is measured from the release point. Each displacement has uncertainty ±0.005 m and each time has uncertainty ±0.005 s. Assume the model y = 1/2 gt².
a) Use the t = 0.400 s reading to estimate g.
b) Estimate the percentage uncertainty in g using percentage uncertainty in y plus twice the percentage uncertainty in t. Give an absolute uncertainty.
c) State whether the result is consistent with g = 9.81 m s⁻².
d) Explain why later frames usually give a more precise estimate than the earliest non-zero frame.
e) Identify two limitations of this video method or of the stated model.
| Time / s | Downward displacement / m |
|---|---|
| 0.000 | 0.000 |
| 0.100 | 0.051 |
| 0.200 | 0.194 |
| 0.300 | 0.447 |
| 0.400 | 0.780 |
Worked answer
Solution
- Step 1
At 0.400 s,
g = 2y / t² = 2(0.780) / 0.400² = 9.75 m s⁻². - Step 2
Percentage uncertainty from displacement:
0.005 / 0.780 × 100% = 0.64%. - Step 3
Time contribution:
2(0.005 / 0.400) × 100% = 2.50%. Total approximate percentage uncertainty:3.14%. - Step 4
Absolute uncertainty:
0.0314 × 9.75 = 0.31 m s⁻², sog ≈ (9.8 ± 0.3) m s⁻². - Step 5
The interval 9.5–10.1 m s⁻² contains 9.81 m s⁻², so the estimate is consistent with the stated value.
- Step 6
Later frames have larger t and y, making the fixed absolute timing and position uncertainties smaller percentages of the measured quantities.
- Step 7
Valid limitations include uncertainty in the exact release frame, parallax or scale calibration, camera perspective, frame-rate timing error, initial push, or fluid resistance making the constant-g model imperfect.
Final answer
g = 9.75 m s⁻² with approximate uncertainty 0.31 m s⁻², reported as (9.8 ± 0.3) m s⁻²; this is consistent with 9.81 m s⁻².
Common mistake
Using only the percentage uncertainty in displacement and forgetting that the time is squared.
Concepts tested
- video analysis
- gravitational acceleration
- percentage uncertainty
- model limitation
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