Question

This question has statement I and statement II. Of the four choices given after the statements, choose the one that best describes the two statements.
Statement - I: A point particle of mass \(m\) moving with speed \(v\) collides with stationary point particle of mass \(M\). If the maximum energy loss possible is given as \(f\left(\frac{1}{2}mv^2\right)\), then \(f = \left(\frac{m}{M+m}\right)\).
Statement - II: Maximum energy loss occurs when the particles get stuck together as a result of the collision.

• A. Statement - I is true, Statement - II is true; Statement - II is the correct explanation of Statement - I.
• B. Statement - I is true, Statement - II is true; Statement - II is not the correct explanation of Statement - I.
• C. Statement - I is true, Statement - II is false.
• D. Statement - I is false, Statement - II true.

Hint:

Maximum kinetic energy is lost in a perfectly inelastic collision (particles stick together). Compute that loss as a fraction of \(\frac{1}{2}mv^2\) and compare with the given \(f\).

Answer 1

The Correct Option is D

Maximum energy loss occurs when the two particles stick together (perfectly inelastic collision), so Statement - II is true.
Momentum conservation: \(mv = (M+m)V\), so \(V = \frac{mv}{M+m}\).
Final KE \(= \frac{1}{2}(M+m)V^2 = \frac{1}{2}\frac{m^2v^2}{M+m}\).
Energy loss \(= \frac{1}{2}mv^2 - \frac{1}{2}\frac{m^2v^2}{M+m} = \frac{1}{2}mv^2\left(1 - \frac{m}{M+m}\right) = \frac{1}{2}mv^2\cdot\frac{M}{M+m}\).
Hence \(f = \frac{M}{M+m}\), not \(\frac{m}{M+m}\), so Statement - I is false.
Answer: (4) Statement - I is false, Statement - II true.