Two Carts and a Doubtful Energy Loss
On a linear air track, cart A (mass 0.500 kg) is given a gentle push and passes a light gate at 0.600 m/s. It collides with cart B (mass 0.300 kg), which is initially at rest. After the collision, A continues in the same direction at 0.300 m/s and B moves at 0.500 m/s. The velocities are measured with an uncertainty of plus or minus 0.020 m/s. The track is supported at two points 1.20 m apart; one support is 8 mm higher than the other, and the lower support is at the starting end so the carts move up the slope. A student calculates that kinetic energy falls from 0.090 J to 0.060 J and concludes: 'Energy is not conserved in this collision, so the conservation law is only an approximation.' The team must decide whether the data support an inelastic collision or whether the apparatus is responsible.
1. Check momentum conservation before and after the collision, including a simple uncertainty estimate. Does the data support conservation?
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Before: p = 0.500 x 0.600 = 0.300 kg m/s. After: p = 0.500 x 0.300 + 0.300 x 0.500 = 0.150 + 0.150 = 0.300 kg m/s. Uncertainty in each momentum term is m times delta v: 0.500 x 0.020 = 0.010 and 0.300 x 0.020 = 0.006. Combining the two after terms gives about 0.012; before is 0.010. The difference is 0.000 plus or minus 0.016 kg m/s, so momentum is conserved within the data. The equality to three decimal places is partly a coincidence of the chosen numbers, not extra precision.
2. Test the student's conclusion. Calculate the missing energy and compare it with the gravitational energy change if the track is tilted.
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KE before = 0.5 x 0.500 x 0.600^2 = 0.090 J. KE after = 0.5 x 0.500 x 0.300^2 + 0.5 x 0.300 x 0.500^2 = 0.0225 + 0.0375 = 0.060 J. Loss = 0.030 J. The track rises by 8 mm over 1.20 m, a slope of 0.00667. Over the 0.60 m collision section the height change is 4.0 mm. If the carts move up the slope, the gravitational energy gained by the moving system is (0.500 + 0.300) x 9.81 x 0.0040 = 0.031 J, almost exactly the missing kinetic energy. So the data do not yet establish an inelastic collision; the 'lost' energy may have become gravitational potential energy.
3. Recommend what the team should do before repeating the experiment, and state what result would support the inelastic interpretation.
Read the worked answer
Level the track with a spirit level and shims, then check by pushing a cart along it and confirming that its speed stays constant within the timing uncertainty. Repeat the collision with the same carts and gates. If the kinetic-energy loss remains about 0.030 J after levelling, and momentum is still conserved, the collision is inelastic and energy has gone into deformation, sound and thermal energy. If the loss falls to within the uncertainty of zero, the original result was an apparatus artefact. Comparing magnetic bumpers with Velcro would provide a further check because the Velcro case should show a larger, genuine loss.