Air Teaser: Calculate a Spud Gun’s Velocity
By Erine Parker, CFPAI, CFPSD, CFPS, CFPMM, CFPMT, CFPMIP, CFPMMH, CFPMIH, CFPE.
Many people enjoy using a safe spud gun. Shooting a potato at 100 psi can be highly entertaining. However, using compressed air to propel something is sometimes frowned upon. Then the challenge becomes to build a vacuum ping pong gun that doesn’t use an air compressor, like this:
Take a 10foot PVC schedule 40 1.5inch pipe with an ID of 1.59 inches. Use a ping pong ball with a diameter of 1.57 inches. Add a Y or a T connection at one end to attach a vacuum pump. Place a piece of mylar over both ends with a female coupler to hold it in place after putting the ball in the pipe. Attach a strong vacuum pump to the branch and pull a vacuum while aiming upward and away from any object or person. Treat it as a loaded rifle. Twentysix inches of mercury will blow the ball out of the far end of the barrel when you puncture the mylar on the back end, but that’s not impressive. Lower the vacuum to 27 inches of mercury and it is much better. But at 28 inches or higher, the results are unbelievable.
Here is your problem: calculate the velocity of the ball leaving the barrel before it hits the atmosphere in meters or feet per second, and mph or kph.
Given: Ball weight 2.7 grams
Barrel is 10 feet long
Vacuum is 28 inches of mercury (Hg)
See the Solution
The solutions below refer to the Fluid Power Reference Handbook, pages 14 and 60 of the first edition (FPRH), and pages 35 and 83 of the second edition (FPRH 2).
F_{a} = (m * a)

F_{a} = Force _{lb}
m = mass _{lbm/32.2}
a = Acceleration _{ft/sec/sec }

F_{a} = Force _{N}
m = mass _{kg}
a = Acceleration _{m/sec/sec }

Force to Accelerate

Metric

US Customary

F = p * A
28 in. hg / 295.3 = 0.0948 MPa
A = d² x .7854 = 40 mm² x .7854 A = 1,256.64 mm²
F = p x A = 0.0948 MPa x 1,256.64 mm² = 119.129 Newtons
Fa = 119.129 Newtons
Fa = m x a
Fa = 119.129 N
Mass = 2.7/1,000 = .0027 kg
a = 119.129 / .0027 = 44,122.03 m/sec/sec
FPFH p. 14 and FPFH 2 p. 35 have the formula:
a = 2D/t2
a = 44,122.03 m/sec/sec
D = 10 feet / 3.28 = 3.05 meters
t = time in seconds
t² = 2 x 3.05 / 44,122.03 m/sec/sec
t² = .000138252, t = .0118 seconds
FPFH p. 60 and FPFH 2 p. 83 have the formula:
∆V = a x t where ∆V = meters/second, a = acceleration m/sec/sec, and t = time in seconds
∆V = 44,122.03 m/sec/sec x .0118 = 521 meters/second
521 x 60 x 60 / 1,000 = 1875.6 kph

F = p * A
28 in. hg / 2.03 = 13.79 psi
A = d² x .7854 = 15.57² x .7854 A = 1.94 in²
F = p x A = 13.79 psi x 1.94 in² = 26.7 lbs.
Fa = 26.7 lbs.
Fa = m x a
Fa = 26.7 lbs.
Mass = 2.7 grams x 0.0022 = 0.00594 lbs.
0.00594 lbs. / 32.2 = 0.00018 slugs
a = 26.7 lbs. / 0.00018 = 144,737 ft/sec/sec
FPFH p. 14 and FPFH 2 p. 35 have the formula:
a = 2D/t2
a = 144,737 ft/sec/sec
D = 10 feet
t = time in seconds
t² = 2 x 10 ft / 144,737 ft/sec/sec t² = .00013818, t = .0118 seconds
FPFH p. 60 and FPFH 2 p. 83 have the formula:
∆V = a x t where ∆V = feet/second, a = acceleration ft/sec/sec, and t = time in seconds
∆V = 144,737 ft/sec/sec x .0118 = 1708 feet/second
1708 x 60 x 60 / 5280 = 1,165.5 mph
