A potato gun fires a potato horizontally down a half-open cylinder of cross-sectional area A. When the gun is fired, the potato slug is at rest, the volume between the end of the cylinder and the potato is V_{0}, and the pressure of the gas in this volume is P_{0}. The atmospheric pressure is P_{\mathrm{atm}}, where P_{0}>P_{\mathrm{atm}}. The gas in the cylinder is diatomic; this means that C_{\mathrm{v}}=5 R / 2 and C_{\mathrm{p}}=7 R / 2. The potato moves down the cylinder quickly enough that no heat is transferred to the gas. Friction between the potato and the barrel is negligible and no gas leaks around the potato.
The parameters P_{0}, P_{\text {atm }}, V_{0}, and A are fixed, but the overall length L of the barrel may be varied.
a. What is the maximum kinetic energy E_{\max } with which the potato can exit the barrel? Express your answer in terms of P_{0}, P_{\text {atm }}, and V_{0}.
b. What is the length L in this case? Express your answer in terms of P_{0}, P_{\text {atm }}, V_{0}, and A.