An adiabatic cylindrical chamber with a frictionless movable piston has been placed on a smooth horizontal surface as shown. One mole an ideal monotonic gas is enclosed inside the chamber. Mass of the piston is M and mass of the remaining chamber including the gas is 4M. The gas is at atmospheric pressure and temperature. A particle of mass M moving horizontally with speed v, strikes the piston elastically. If change in temperature of the gas when the compression is maximum is $K\times \frac{M{v}^2}{15R}$then K will be_____

Immediately after collision, the particle comes to rest and the piston starts moving with velocity v, because the collision is elastic and two colliding objects have equal mass. The compression is maximum when the piston stops moving relative to the cylinder. Let $v_0=$ velocity of entire system at the point of maximum compression The first figure shows situation immediately after collision and the second figure shows situation at maximum compression. 

Momentum conservation gives-

$5M{v}_0=Mv\Rightarrow v_0=\frac{v}{5}$

Energy conservation gives-

$\begin{array}{rr} & n{C}_v\mathrm{\Delta }T=\frac{1}{2}M{v}^2-\frac{1}{2}\left(5M\right){\left(\frac{V}{5}\right)}^2\\ & 1\times \frac{3}{2}R\mathrm{\Delta }T=\frac{2}{5}M{v}^2\left[\text{ for monatomic gas }{C}_v=\frac{3}{2}R\right]\\ & \mathrm{\Delta }T=\frac{4M{v}^2}{15R}\end{array}$