Show that, in one-dimensional flow of a perfect gas at constant entropy, the Riemann invariants $u \pm 2\left(c-c_{0}\right) /(\gamma-1)$ are constant along characteristics $d x / d t=u \pm c$.

A perfect gas occupies a tube that lies parallel to the $x$-axis. The gas is initially at rest and is in $x>0$. For times $t>0$ a piston is pulled out of the gas so that its position at time $t$ is

$$x=X(t)=-\frac{1}{2} f t^{2}$$

where $f>0$ is a constant. Sketch the characteristics of the resulting motion in the $(x, t)$ plane and explain why no shock forms in the gas.

Calculate the pressure exerted by the gas on the piston for times $t>0$, and show that at a finite time $t_{v}$ a vacuum forms. What is the speed of the piston at $t=t_{v}$ ?