Two identical particles of rest mass m₀ are each moving toward the other with speed u in frame S. The particles collide inelastically with a spring that locks shut (Figure) and come to rest in S, and their initial kinetic energy is transformed into potential energy. In this problem you are going to show that the conservation of momentum in reference frame S’, in which one of the particles is initially at rest, requires that the total rest mass of the system after the collision be 2m₀/√1 - u²/c².

 (a) Show that the speed of the particle not at rest in frame S’ is u’ = 2u/(1 + u² / c²) and use this result to show that

 (b) Show that the initial momentum in frame S’ is p’ = 2m₀u/(1 - u²/c²).

(c) After the collision, the composite particle moves with speed u in S’ (since it is at rest in S). Write the total momentum after the collision in terms of the final rest mass M₀, and show that the conservation of momentum implies that M = 2m₀/√1 – u²/c².

(d) Show that the total energy is conserved in each reference frame.

Transcribed Image Text:

(a) m (b)