Momentum on a Misleveled Air Track
During a collision lab, a student uses a 1.20 m air track that is not level: one end is 3.0 cm higher than the other, although the bubble level was misread as centered. Glider A (0.482 kg) is launched uphill toward stationary glider B (0.716 kg). A photogate records A’s speed as 0.62 m/s before the collision. After the spring bumpers interact, A moves back downhill at 0.18 m/s and B moves uphill at 0.41 m/s. The student calculates only the magnitudes and concludes that momentum is not conserved, so the bumpers must be faulty. The collision occurs 0.30 s after A passes the first gate, and B reaches the second gate 0.08 s after the collision. The track’s air supply is weak, and the student did not calibrate the photogate spacing. The class average for this experiment is usually within 3% of momentum conservation.
1. Using the given masses and velocities, test the student’s claim. Include the direction of the external impulse from gravity along the track and state whether the data are consistent with momentum conservation.
Read the worked answer
Take uphill as positive. The initial momentum is
(p_i=(0.482)(0.62)=0.299\ \mathrm{kg,m/s}).
After the collision,
(p_f=(0.482)(-0.18)+(0.716)(0.41))
(p_f=-0.0868+0.294=0.207\ \mathrm{kg,m/s}).
Thus (\Delta p=-0.092\ \mathrm{kg,m/s}) (downhill). The track slope is (\sin\theta=0.030/1.20=0.025), so the downhill acceleration component is (g\sin\theta\approx0.245\ \mathrm{m/s^2}). Over about (0.38\ \mathrm{s}), the external impulse from gravity is approximately
(J_{\mathrm{ext}}=-(1.198)(0.245)(0.38)\approx-0.11\ \mathrm{kg,m/s}).
This has the right sign and is comparable to the observed (-0.092\ \mathrm{kg,m/s}). The apparent nonconservation is therefore consistent with the unlevel track, not with faulty bumpers. A level track is needed before claiming a violation.
2. The student also finds that the total kinetic energy drops from 0.093 J to 0.068 J. Does this prove the collision is inelastic? Explain what else must be accounted for and what follow-up measurement would settle it.
Read the worked answer
The kinetic-energy loss is real for these data: (K_i=\tfrac12(0.482)(0.62)^2=0.093\ \mathrm{J}), and (K_f=\tfrac12(0.482)(0.18)^2+\tfrac12(0.716)(0.41)^2=0.068\ \mathrm{J}), a loss of about 0.025 J. But this alone does not prove the bumpers are inelastic, because the gliders change height on the tilted track and gravity does work. One must compare (K+U_g) before and after, not (K) alone. Air-track friction and spring-bumper energy storage also matter. A follow-up measurement should level the track, record each glider’s height or use a level track, and use two photogates close to the collision so external impulses are small. Then a remaining mechanical-energy loss would support an inelastic collision.