3. In ICF (inertial confinement fusion), a pellet (shell) filled with deuterium-tritium mixture is compressed by shaped laser pulse. The goal is to achieve the maximum possible density at the peak shell convergence. The relation between pressure and density for a perfect gas, p = Po (S-50)/cy shows that for a given drive pressure provided by the laser, shell density is at maximum if entropy s is constant during the compression (adiabatic compression). Drive pressure scales with the laser intensity. Hence, laser pulse needs to be shaped to insure adiabatic shell compression throughout the implosion. As a model of an ICF implosion, consider a tube of length R filled with perfect gas of uniform density po at pressure po. The tube is closed at the side x=R (this represents center of the pellet). Gas inside the tube is compressed by a piston moving inside the tube. The initial position of the piston is X(0)=0. Determine piston trajectory X(t) required for isentropic gas compression (without shock formation). Calculate pressure evolution at the piston position.