A perfect gas occupies a tube that lies parallel to the $x$-axis. The gas is initially at rest, with density $\rho_{1}$, pressure $p_{1}$ and specific heat ratio $\gamma$, and occupies the region $x>0$. For times $t>0$ a piston, initially at $x=0$, is pushed into the gas at a constant speed $V$. A shock wave propagates at constant speed $U$ into the undisturbed gas ahead of the piston. Show that the pressure in the gas next to the piston, $p_{2}$, is given by the expression

$$V^{2}=\frac{\left(p_{2}-p_{1}\right)^{2}}{\rho_{1}\left(\frac{\gamma+1}{2} p_{2}+\frac{\gamma-1}{2} p_{1}\right)} .$$

[You may assume that the internal energy per unit mass of perfect gas is given by

$$E=\frac{1}{\gamma-1} \frac{p}{\rho}$$