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Chapter 5: Problem 151

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 \(f\left(\frac{1}{2} m v^{2}\right)\) 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 not a correct explanation of Statement-I. (B) Statement-I is true, Statement-II is false. (C) Statement-I is false, Statement-II is true. (D) Statement-I is true, Statement-II is true, Statement-II is a correct explanation of Statement-I.

Short Answer

Expert verified
(D) Statement-I is true, Statement-II is true, Statement-II is a correct explanation of Statement-I.

Step by step solution

01

Initial conditions

Initially, we have a point particle of mass M moving with a speed v, and a stationary point particle of mass m. Considering total mechanical energy of the system: \(E_{initial}=\frac{1}{2} Mv^2 + 0= \frac{1}{2} Mv^2\)
02

Final conditions

After the collision, we assume maximum energy loss occurs when both particles stick together moving with the same final speed V.
03

Conservation of momentum

As no external forces are acting on the two-particle system, we will have conservation of momentum: \(Mv + m \cdot 0 = (M + m)V\) From this equation, we get the final velocity of the particles after the collision: \(V = \frac{Mv}{M + m}\)
04

Final energy of the system

The final total mechanical energy of the system is given by: \(E_{final} = \frac{1}{2}(M+m)V^2 = \frac{1}{2}(M+m)\left(\frac{Mv}{M + m}\right)^2\)
05

Calculate the energy loss

To calculate energy loss, we subtract the final energy from the initial energy: \(Energy Loss= E_{initial} - E_{final} = \frac{1}{2} Mv^2 - \frac{1}{2}(M+m)\left(\frac{Mv}{M + m}\right)^2\) Simplifying the expression, we get: \(Energy Loss =\frac{1}{2}mv^2\frac{m}{M+m}\) Comparing this result to Statement-I, we get the function for maximum energy loss: \(f = \left(\frac{m}{M+m}\right)\) Thus, Statement-I is true.
06

Evaluate Statement-II

The approach we took in our solution was to assume that maximum energy loss occurs when the particles stick together after collision, which led to the correct result for Statement-I. So, Statement-II is true and it is a correct explanation for Statement-I. Answer: (D) Statement-I is true, Statement-II is true, Statement-II is a correct explanation of Statement-I.

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