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.

$\text{Statement-I}:$ 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 $f\left(\frac{1}{2}m{v}^2\right)\text{then}f=\left(\frac{m}{M+m}\right)$.

$\text{Statement-II}:$ 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 false, Statement-II is true.
• C. Statement I is true, Statement-II is false.
• D. Statement-I is true, Statement-II is true, Statement-II is not the correct explanation of Statement.

Answer 1

The correct option is B Statement-I is false, Statement-II is true.

The maximum Energy loss, $E=\frac{p^2}{2m}$
where,
$p$ = momentum,
$m$ = the mass moving of the particle.

Maximum energy loss occurs when the particles get stuck together as a result of the collision.

Maximum energy loss,

$\Delta E=\frac{P^2}{2m}-\frac{P^2}{2(m+M)}$

Where, $(m+M)=$ resultant mass when the particle get stuck.

$\Delta E=\frac{P^2}{2m}[1-\frac{m}{(m+M)}]$
$=\frac{P^2}{2m}\left[\frac{M}{(m+M)}\right]$

Also, $P=mv$

$\Delta E=\frac{m^2{v}^2}{2m}\left[\frac{M}{(m+M)}\right]=\frac{m{v}^2}{2}\left[\frac{M}{(m+M)}\right]$

Comparing the expression with$f\left(\frac{1}{2}m{v}^2\right)$. Then,

$f=\frac{M}{m+M}$

Hence, Statement-I is wrong and statement-II is correct. So, the correct answer is $\left(D\right)$.