The Bounce That Appeared to Gain Energy
Students drop a tennis ball from a measured height of 1.50 m and record the bounce with a smartphone at 240 frames per second. They track the ball's center frame by frame and compute a rebound height of 0.92 m, giving a coefficient of restitution e = √(0.92/1.50) = 0.78. A second group, using the same ball and height, obtains 0.98 m and e = 0.81. A third group reports that in one frame the ball's center appears above the release height, which they jokingly call 'free energy.' The teacher sees that the meter stick is clamped to the back edge of the lab table, about 0.35 m behind the ball's vertical path; the phone is 2.0 m away and slightly above the ball; the ball is blurred at contact; and the phone uses a rolling shutter. The class must decide which rebound height to report and how to revise the experiment before writing a lab conclusion.
1. Explain why the frame that appears to show the ball above its release height is almost certainly an artifact. Identify at least three specific features of the setup that could produce this apparent result.
Read the worked answer
A ball cannot rebound above its release height without an external energy input; the gravitational potential energy at release is the maximum available (minus losses). The apparent 'above release' frame therefore points to a measurement artifact, not free energy.
Specific causes:
- Parallax: the meter stick is 0.35 m behind the ball's plane. The camera is 2.0 m away and slightly above. When the ball is at different heights, its line of sight to the camera crosses the stick at a different point, so the ball's image can appear higher or lower relative to the stick than its true height.
- Rolling shutter: the phone's sensor is read row by row over a finite time. A fast-moving ball can be captured at slightly different times in different rows, skewing or stretching the image, especially near contact.
- Motion blur at contact: the ball's center is poorly defined in the blurred frame, so the tracking software may pick a point on the blur streak rather than the true center.
- Perspective and lens distortion: the phone is slightly above the ball and not perfectly perpendicular to the plane of motion; straight lines can project nonlinearly.
- Frame timing: the frame labelled as release or apex may not correspond exactly to the event, and the ball's center at release may be estimated from a blurred frame.
2. Which rebound height should the class report, and how should they combine the three groups' data? Justify your answer in terms of random and systematic error.
Read the worked answer
They should not report the raw average of 0.92 m, 0.98 m, and the 'above release' frame. The above-release frame should be excluded after diagnosing it as an artifact; it is an outlier caused by a known systematic effect, not a valid measurement. The remaining 0.92 m and 0.98 m differ by 6 cm, which is plausibly random scatter from tracking and frame timing.
However, both values share the same systematic parallax because the stick is behind the ball. The class should first correct the scale by placing a reference length in the ball's plane, or by calibrating the apparent height using a stationary object at the ball's distance. After correction, they can combine the corrected means from each group with a weighted mean, using the scatter within each group's repeated drops to estimate random uncertainty. The reported result should be a corrected height with an uncertainty, not a single bare number. If the correction is not possible, they should report the value with a statement that parallax makes the absolute height uncertain by at least the size of the discrepancy.
3. Describe an improved design that would reduce both random and systematic uncertainty in the coefficient of restitution.
Read the worked answer
Put a vertical meter stick or calibration grid in the same plane as the ball's path, so parallax is removed. Fix the camera on a tripod, perpendicular to that plane, at the height of the bounce, and keep the distance fixed. Use the highest frame rate available (for example 960 fps or a motion sensor) and diffuse, bright lighting; use a ball with a clear pattern or a small marker so the center can be tracked. Record several drops from the same height, and measure both the release height and the rebound height consistently-e.g., bottom of the ball to the floor-using subpixel tracking software. Use a hard, level surface and the same ball. Compute e from the heights, e = √(h_rebound/h_release), and propagate the reading uncertainty. To separate systematic from random error, repeat with the calibration stick in two positions and compare; if the result shifts, the correction is working.