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A particle of mass m with kinetic energy K and momentum p collides head-on elastically with a particle of mass 2m at rest. Match the following quantities after the collision: List-I: (P) Magnitude of momentum of first particle (Q) Magnitude of momentum of second particle (R) Kinetic energy of first particle (S) Kinetic energy of second particle List-II: (1) 3p/4 (2) K/9 (3) p/3 (4) 8K/9 (5) None of these

1. P->1; Q->5; R->4; S->3
2. P->2; Q->3; R->4; S->1
3. P->3; Q->5; R->2; S->4
4. P->1; Q->2; R->3; S->5

Correct answer: P->3; Q->5; R->2; S->4

Solution

Let initial velocity of mass m be u. Elastic collision formulas: v1' = (m1-m2)/(m1+m2)*u = (m-2m)/(3m)*u = -u/3. v2' = 2m1/(m1+m2)*u = 2m/(3m)*u = 2u/3. P (|momentum of m after|): m*|v1'| = m*u/3 = p/3 -> matches (3). Q (|momentum of 2m after|): 2m*v2' = 2m*2u/3 = 4mu/3 = 4p/3. This is not in the list (not 3p/4, not p/3) -> matches (5) None of these. R (KE of m after): (1/2)m*(u/3)² = (1/2)mu²/9 = K/9 -> matches (2). S (KE of 2m after): (1/2)(2m)(2u/3)² = m*4u²/9 = 4mu²/9 = 4*2K/9 = 8K/9 -> matches (4). Matching: P->3, Q->5, R->2, S->4.

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