Cart on a Ramp: Is the Missing Energy Small Enough to Ignore?
A 0.50 kg cart is released from rest at a height of 0.40 m above the bottom of a ramp. A photogate at the bottom records the cart's speed. The theoretical prediction, ignoring friction, is v = sqrt(2gh) = 2.8 m/s. Five trials give speeds of 2.4, 2.3, 2.5, 2.4, and 2.4 m/s. The cart's wheels are small and hard, and the ramp is smooth but not polished. A student says the missing energy can be ignored because the average speed, 2.4 m/s, is 'close enough' to 2.8 m/s. Another student claims the difference is caused by air resistance. The teacher asks the group to quantify the discrepancy, decide whether friction is negligible at the precision of the experiment, and design a follow-up test that could distinguish between friction and a systematic timing error.
1. Calculate the initial gravitational potential energy, the final kinetic energy using the average speed, and the percentage of mechanical energy not recovered. Does the result support the 'close enough' claim?
Read the worked answer
Initial PE = mgh = (0.50 kg)(9.8 m/s²)(0.40 m) = 1.96 J. Using the mean speed 2.4 m/s, final KE = ½(0.50)(2.4)² = 1.44 J. The missing mechanical energy is 1.96 − 1.44 = 0.52 J, or about 27% of the initial PE. A 27% loss is not negligible at this precision; the 0.4 m/s speed difference is about 14%, and squaring it doubles the relative effect in energy. The student's 'close enough' judgment would need a stated tolerance. If the experiment is intended to test conservation, this result fails the test unless friction is accounted for.
2. Evaluate the air-resistance claim and describe a test to distinguish friction from a timing error.
Read the worked answer
Air resistance is unlikely to be the main cause here: at 2–3 m/s, drag on a small cart is much smaller than the observed 0.52 J loss, and the cart's wheels and axle are more plausible friction sources. To distinguish friction from timing error, use two photogates a measured distance apart and compare average speed over that interval with the single-gate speed. If the single gate is systematically off, the two methods will disagree in a consistent ratio. To test friction, run the same cart on a smooth track versus a rougher surface, or add mass to the cart; friction should change with normal force and surface, while a timing offset should not. A motion sensor could also produce a position-time graph to check speed independently.
3. The group wants to improve the experiment. Propose two changes and explain what each would control or reveal.
Read the worked answer
One change is to use an air track or low-friction cart and measure speed with two gates; this reduces wheel friction and checks the photogate timing. A second change is to add known masses to the cart and repeat at the same release height; if the lost energy is frictional, the fraction lost should change as normal force changes, whereas a pure timing error would stay proportional. The group should also measure release height with a meter stick and repeat several times to get a mean and spread. These changes separate random scatter from systematic effects.