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

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