Trading Momenta In A Collision
Trading Momenta in a Collision?
Two particles move perpendicular to each other until they collide. Particle 1 has mass m and momentum of magnitude 2p, and particle 2 has mass 2m and momentum of magnitude p.
Suppose that after the collision, the particles “trade” their momenta, as shown in the figure. That is, particle 1 now has magnitude of momentum p, and particle 2 has magnitude of momentum 2p; furthermore, each particle is now moving in the direction in which the other had been moving. How much kinetic energy, K_lost, is lost in the collision?
Express your answer in terms of m and p.
Consider an alternative situation: This time the particles collide completely inelastically. How much kinetic energy K_lost is lost in this case?
Energy of particle:
E = 1/2 p²/m
E1 = 1/2 [(2p)²/m + p²/(2m)] = 1/2 x 9/2 x p²/m
E2 = 1/2 [p²/m + (2p)²/(2m)] = 1/2 x 3 x p²/m
Answer(a)
Energy loss E1 – E2 = 3/4 p²/m
Net momentum (Pithagorean):
Po² = (2p)² + (p)² = 5p²
Net mass
M = m + 2m = 3m
Energy of the center of mass:
Ecm = 1/2 Po²/M = 1/2 x 5/3 x p²/m
Answer(b)
Maximum energy loss possible is
E1 – Ecm = 1/2 [9/2 - 5/3] p²/m = 17/12 p²/m
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