Question

You and your younger brother are designing an air rifle that will shoot a lead pellet with mass m = 1.00 g and cross-sectional area A = 0.0150 cm2. The rifle works by allowing high-pressure air to expand, propelling the pellet down the rifle barrel. Because this process happens very quickly, no appreciable thermal conduction occurs and the expansion is essentially adiabatic. Your design is such that, once the pressure begins pushing on the pellet, it moves a distance of L = 40.0 cm before leaving the open end of the rifle at your desired speed of v = 100 m/s. Your design also includes a chamber of volume V = 15.0 cm3 in which the high-pressure air is stored until it is released. Your brother reminds you that you need to purchase a pump to pressurize the chamber. To determine what kind of pump to buy, you need to find what the pressure of the air must be in the chamber (in Pa) to achieve your desired muzzle speed. Ignore the effects of the air in front of the bullet and friction with the inside walls of the barrel. (Assume that air is diatomic.) _________ Pa

Answer 1

The kinetic energy of bullet is

E = 1/2 * m * v²

E = 1/2 * 1e-3 * 100²

E = 5 J

Therefore, work done on the gas is -5 J

Now, use equations for adiabatic expansion

W = (1 / $\gamma$ - 1) ( P_fV_f - P_i V_i) --------------- (1)

Here,

V_f = 15 cm³ + 0.015 * 40

V_f = 15.6 cm³

$\gamma$ = 1.4 (diatomic gas)

so,

Also

P_f = P_i (V_i / V_f)^$\gamma$

Put this is (1) and solve for initial pressure , P_i

P_i = 8.565e6 Pa