Problem

Compressed gas is used in many circumstances to propel objects within tubes. For example,...

Compressed gas is used in many circumstances to propel objects within tubes. For example, you can still find pneumatic tubes in use in many banks to send and receive items to drive-through tellers,* and it is compressed gas that propels a bullet out of a gun barrel. Let the cross-sectional area of the tube be given by A and the position of the cylinder by s, and assume that the compressed gas is an ideal gas at constant temperature so that the pressure P times the volume Ω is a constant, i.e., PΩ = constant. Show that the potential energy of this compressed gas is given by V = −P0s0A ln(s/s0), where P0 is the initial pressure and s0 is the initial value of s. Model the cylinder as a particle, and assume that the forces resisting the motion of the cylinder are negligible.