Question

An air gun fires a bullet by means of compressed air. Simplistically, an air gun can be viewed as a
metal cylindrical tube open on one end and plugged air-tight in the middle by the bullet, which
is held in place by a trigger. The space between the bullet and the blind end is filled with
compressed air. When the trigger is released, the compressed air undergoes a sudden expansion
against ambient pressure, pushing the bullet out of the tube at high speed.

 Pressure of compressed air = 11.0 bar
 Temperature of compressed air = 298 K
 Mass of bullet = 5 g
 Length of metal tube = 0.9 m
 Initial position of the bullet = 0.15 m from the blind end
 Cross-sectional area of tube = 0.4 cm2

Ambient condition is 298 K and 1 atm. Constant-volume heat capacity of air = 2.5R, where R is the gas constant.

(a) If we fire this toy air gun in a horizontal direction, 1.5 m from the ground, what is the
theoretical upper limit for the horizontal displacement of the bullet before it falls to the
ground, given that the expansion is so quick that no heat transfer can occur? Hint: To obtain
the maximum amount of work, you should assume that the compressed air expansion is
quasi-static. This is unrealistic but will give us the theoretical upper limit.

(b) Suppose we can lengthen or shorten the gun at the open end. Find the optimal gun length
that will maximize the horizontal displacement of the bullet in normal operation, under the
assumption of quasi-static and adiabatic expansion. Can you offer a physical explanation why
the bullet will not fly as far if the gun is longer or shorter than this optimal length?

Clarification: A comment mentioned the 1.5m from the ground. This means that the gun is at an elevation of 1.5m.

Answer 1