A particle of mass m, kinetic energy K and momentum. p collides head on elastically with another particle of mass 2 m at rest. Match the following columns (after collision) and mark the correct option from the codes given below.
Column I Column II
(A) Momentum of first particle (p) 4p/3
(B) Momentum of second particle (q) K/9
(C) Kinetic energy of first particle (r) $-$ p/3
(D) Kinetic energy of second particle (s) 8K/9
A B C D

Moderate

Solution

$p_{1} + p_{2} = p$ …(i)
Further, $K_{1} + K_{2} = K$
or $\frac{p_{1}^{2}}{2 m} + \frac{p_{2}^{2}}{4 m} = \frac{p^{2}}{2 m}$ or $2 p_{1}^{2} + p_{2}^{2} = 2 p^{2}$ …(ii)
Solving Eqs. (i) and (ii), we get
$p_{2} = \frac{4}{3} p$ and $p_{1} = - \frac{p}{3}, K_{1} = \frac{K}{9} and K_{2} = \frac{8 K}{9}$
Hence, $A \to r, B \to p, C \to q, D \to s .$

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Column I	Column II
(A) Momentum of first particle	(p) 4p/3
(B) Momentum of second particle	(q) K/9
(C) Kinetic energy of first particle	(r) $-$ p/3
(D) Kinetic energy of second particle	(s) 8K/9