$(6$ points$)2.$ Consider the following shock tube problem:

$p_L=1\mathrm{.}0,{}_L=1\mathrm{.}0,u_L=0\mathrm{.}0$ and $p_R=0\mathrm{.}04379,{}_R=0\mathrm{.}1,u_R=0\mathrm{.}0$ Also, $\frac{p_2}{p_1}=5\mathrm{.}312.$ Pressures are in bar and densities are in $k\frac{g}{m^3}.$ All required calculations below must be done by hand, not with Matlab. However, you may use the provided $q,$ that calculates the exact solution of the shock tube problem $($Sod$\text{'}$s problem$)$ to check your answers.

a$.$ Compute the induced velocity $u_p$ behind the shock.

b$.$ Use these values and for time $t=0\mathrm{.}05sec$ to calculate the location of the shock, the contact, and the tail and head of the expansion fan.