Question

A horizontal piston cylinder arrangement encloses a gas at a pressure $P_0$. The atmospheric pressure is also $P_0$. Now, a particle of mass $m$ strikes the piston of identical mass $m$ with a horizontal speed $u$ and sticks to it. Subsequently, the piston moves without friction and compresses the gas. The process of compression can be assumed to be adiabatic. In this process

• A. When the piston comes to rest at maximum compression, the internal energy of gas would be increased by $\frac{1}{4}{\text{mu}}^2$.
• B. During the time interval starting from the mass striking the piston to the maximum compression of the enclosed gas, the work done by the gas is $-\frac{1}{4}{\text{mu}}^2$
• C. When the piston comes to rest after compressing the gas, the internal energy of the gas would have increased by the amount more than $\frac{1}{4}{\text{mu}}^2$.
• D. During the compression of the gas till the time when piston comes to rest, work done by the gas is less than $-\frac{1}{4}{\text{mu}}^2$

Answer 1

Answer: (C), (D)

During the compression of the gas till the time when piston comes to rest, work done by the gas is less than $-\frac{1}{4}{\text{mu}}^2$

Applying work energy theorem:
$W_{\text{gas}}+W_{\text{atm}}=\Delta KE=-\frac{1}{4}m{u}^2$
$\Rightarrow W_{\text{gas}}=-\frac{1}{4}m{u}^2-W_{atm}=-\Delta U_{\text{gas}}$
Since, the gas is compressed
$\therefore W_{\text{atm}}>0$
$\Rightarrow W_{\text{gas}}<-\frac{1}{4}m{u}^2$
and $\Delta U_{\text{gas}}>\frac{1}{4}m{u}^2$