Spring to propel object at specific velocity

[Becky6]

I want to make a small spring piston to propel a 3 mm stainless steel ball to hit a target at a velocity of 15 m/s from just a few inches away. How can I calculate/determine the right spring to use?

 

[HallsofIvy]

Well, you ought to be able to do this using "conservation of energy". If the spring has spring constant k, and you compress it a distance x, it gains potential energy (1/2)kx². When it is released, that potential energy goes into the kinetic energy of the ball, (1/2)mv²: (1/2)kx²= (1/2)mv² so the speed with which the ball leaves the spring is $v= x\sqrt{(k/m)}$. I don't think the ball will lose much speed in "just a few inches". The density of steel varies a little depending on the kind of steel- I'll let you look that up.

 

[Bob S]

Here is a site giving correct potential energy of spring:
http://www.ajdesigner.com/phphookeslaw/spring_potential_energy_equation.php

 

[HallsofIvy]

Oh, blast- I knew better than that!
The potential energy is $(1/2)kx^2$, not $(1/2)kx$. That doesn't even have the correct units!

Thanks for the correction, Bob S, I have edited my post to correct that.

 

[Becky6]

Thank you both so much! That makes sense. With your help I was able to figure out what I need.