Compressed gas is used in many circumstances to propel objects within tubes. 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 is a constant, i.e., P = constant. If the force exerted by compressed gas is assumed to be a conservat ive force, show that the potential energy of this compressed gas is given by V = - P 0 S 0 A · ln( s / s 0 ) , where P 0 is the initial pressure and s 0 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.