An air rifle shoots a lead pellet by allowing high-pressure air to expand, prope

Question

An air rifle shoots a lead pellet 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. Suppose the rifle starts with 14.5 cm3 of compressed air, which behaves as an ideal gas with ? = 1.40. The expanding air pushes a 1.05-g pellet as a piston with cross-sectional area 0.0400 cm2 along the 59.0-cm-long gun barrel. What initial pressure is required to eject the pellet with a muzzle speed of 110 m/s? Ignore the effects of the air in front of the bullet and friction with the inside walls of the barrel. (atm)

Explanation / Answer

mass of the pellet   m = 1.05 x 10-3 kg

cross sectional area of piston A = 0.0400 cm2  

intial volume of compressed air Vi = 14.5 cm3   

speed of the bullet v = 110 m/s

length of barall     L = 59.0 cm

W = 1/2 m v2 = 0.5 * 1.05 x 10-3 * 1102 = 6.35 J

Pf Vf = Pi Vi

1/Y-1 (Pf Vf - Pi Vi) = -6.35

1/1.4-1 ((PiVi/Vf) Vf - Pi Vi) = -6.35

Vf = 14.5 + 59.0 cm (0.0400) = 14.52 cm3

Pi= (-6.35 * 0.4 * 106) / (14.52 * (14.5 / 14.52)1.4 - 14.5 )

= 3.178 x 108 Pa = 3136 atm

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