$K . E_{i}+P . E_{i}=K . E_{f}+P . E_{f}$
$\Rightarrow 0+m g h=\frac{1}{2} m v^{2}$
$\Rightarrow v=\sqrt{2 g h}$
Let the common velocity of both block and putty be $v^{\prime}$
At collision momentum balance:
$m v=m v^{\prime}+m v^{\prime}$
$\Rightarrow \sqrt{2 g h}=2 v^{\prime}$
$\Rightarrow v^{\prime}=\sqrt{\frac{g h}{2}}$
Again applying energy conservation (both block and putty at max height)
$K . E_{i}+P . E_{i}=K . E_{f}+P . E_{f}$
$\Rightarrow \frac{1}{2} 2 m v^{\prime 2}=2 m g h^{\prime}$
$\Rightarrow \frac{1}{2} \frac{g h}{2}=g h^{\prime}$
$\Rightarrow h^{\prime}=h / 4$
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K. The heat capacities per mole of an ideal monatomic gas are $C_v=\frac{3}{2} \ R, C_p=\frac{5}{2} R$, and those for an ideal diatomic gas are $C _{ v }=\frac{5}{2} \ R , C _{ P }=\frac{7}{2} R$.
$1.$ Consider the partition to be rigidly fixed so that it does not move. When equilibrium is achieved, the final temperature of the gases will be :
$(A)$ $550 \ K$ $(B)$ $525 \ K$ $(C)$ $513 \ K$ $(D)$ $490 \ K$
$2.$ Now consider the partition to be free to move without friction so that the pressure of gases in both compartments is the same. Then total work done by the gases till the time they achieve equilibrium will be:
$(A)$ $250 \ R$ $(B)$ $200 \ R$ $(C)$ $100 \ R$ $(D)$ $-100 \ R$
Give the answer question $1$ and $2.$