Air-track collision: momentum conservation and systematic tilt
Students use a level air track to test momentum conservation in a collision. Glider A (mass 0.250 kg) passes a light gate at 0.640 m/s and collides with glider B (mass 0.150 kg), which is initially at rest. The gliders stick together with Velcro and pass a second gate at 0.380 m/s. The track is nominally level, but a spirit level shows a slight tilt; the students did not adjust the air flow, and the second gate is 0.45 m beyond the collision point. The accepted value for momentum is taken from the before-collision measurement. The students calculate momentum before and after, and find a 5% difference. They must decide whether the data support conservation of momentum within experimental uncertainty. The speed values have an uncertainty of ±0.010 m/s.
1. Calculate the total momentum before and after the collision, the percentage difference, and the kinetic energy lost. Compare the kinetic energy loss with the value expected for a perfectly inelastic collision.
Read the worked answer
Before:
p_before = 0.250 × 0.640 = 0.160 kg m/s
After: the combined mass is 0.250 + 0.150 = 0.400 kg, so
p_after = 0.400 × 0.380 = 0.152 kg m/s
Difference:
0.160 − 0.152 = 0.008 kg m/s
percentage difference = 0.008 / 0.160 × 100 = 5.0%
Kinetic energy before:
KE_before = 0.5 × 0.250 × 0.640² = 0.0512 J
Kinetic energy after:
KE_after = 0.5 × 0.400 × 0.380² = 0.0289 J
KE_loss = 0.0512 − 0.0289 = 0.0223 J
For a perfectly inelastic collision, momentum conservation would give
v = 0.160 / 0.400 = 0.400 m/s
and KE_after = 0.5 × 0.400 × 0.400² = 0.0320 J, so the expected loss is 0.0192 J. The measured loss, 0.0223 J, is larger than expected, which is consistent with the measured final speed being lower than 0.400 m/s.
2. Assess whether the 5% difference is significant given the ±0.010 m/s speed uncertainty. What systematic effect could explain the direction of the difference?
Read the worked answer
The momentum uncertainty from the before speed is
δp_before = 0.250 × 0.010 = 0.0025 kg m/s
and from the after speed is
δp_after = 0.400 × 0.010 = 0.0040 kg m/s
Combining in quadrature gives about
δp_total = sqrt(0.0025² + 0.0040²) = 0.0047 kg m/s
The observed difference is 0.0080 kg m/s, about 1.7 times this combined uncertainty. That is marginal: the data do not show a large violation, but the difference is not comfortably within the stated random uncertainty.
The direction matters: the final momentum is lower than the initial momentum. That points to a net external impulse opposite to the motion. A slight upward tilt of the track along the direction of travel would give a gravity component opposing motion, and air resistance or friction would do the same. Either could remove momentum between the gates. A tilt in the opposite direction would have made the final momentum too high, so the sign of the discrepancy is useful evidence for a systematic effect rather than random scatter.
3. Explain why kinetic energy is not expected to be conserved in this collision, and whether the missing kinetic energy invalidates the momentum test.
Read the worked answer
The gliders stick together, so the collision is perfectly inelastic. Kinetic energy is converted into deformation of the Velcro and glider surfaces, sound, and internal thermal energy. Kinetic energy is therefore expected to fall, and the measured 43% loss is not by itself a failure of the experiment.
Momentum, unlike kinetic energy, is conserved when the net external impulse is negligible. During the short collision, the external forces along the track (gravity component from the tilt, friction, air resistance) are small compared with the collision force, so momentum should be approximately conserved. The kinetic energy loss does not invalidate the momentum test; it is a consequence of the inelastic nature of the collision.
However, the energy loss can be used as a check. For a perfectly inelastic collision with the same masses and initial speed, the expected final speed is 0.400 m/s and the expected kinetic energy loss is 0.0192 J. The measured final speed of 0.380 m/s gives a larger loss, which supports the suggestion that an external force also acted along the track. The momentum discrepancy and the excess energy loss are therefore consistent with a small systematic tilt or friction effect.