How to calculate the velocity of a spring loaded projectile

Question

My problem is this. I would like to propel a foam nerf type football that is a little under 2" in diameter and weighs 1.0 oz out of a plastic tube to a velocity of 275 feet per second. I am trying to size a compression spring that will power this. Ideally the spring would have an outer diameter of 1.94" so it will be guided by the inside of the tube. The

spring would be compressed then released to propel the nerf. If I can

calculate the required spring rate I could then size the spring wire diameter, length and coils required. Any thoughts would be appreciated.

Answer 1

Your first equation is to use the law of conservation of energy, i.e., the energy of the football must all come from the energy of the spring.

Ideally,

Energy from spring = Energy absorbed by football

In the actual design, there will be some friction losses but for the purpose of this problem's analysis, friction losses will be disregarded.

Energy from spring = (1/2)(kx^2)

Energy absorbed by football = (1/2)mV^2

where

k = spring constant

x = initial compressed length of the spring

m = mass of the football = 1/(16*32.2)

V = velocity of football

Equating the two energies:

(1/2)(kx^2) = (1/2)(1/16*32.2)(275^2)

kx^2 = 147.71 ft-lbs.

From what you have here (kx^2 = 147.71 ft-lbs.), you can consult any book on Mechanical Engineering Machine Design and read on "Spring Design" and you will have your solution there.

Since this is a design problem, your solution is as good as mine. The final specs of the spring will be dependent on what kind of assumptions will be made.

Good luck.

Answer 2

You want the nerf to leave the tube at 275 ft/s but you don't say how long the spring must be. Have you thought of possibly using a large elastic band. That would be easier and cheaper to use. You know like a giant sling-shot.

Then again, you might want to look into re-designing the potato launcher below: