# What is/is there an equation to figure the speed a projectile travels, based on the PSI or force applied? Answered

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Then why not phrase it that way? "If I have x psi, what is the velocity of a projectile y of z mass and m diameter assuming a path of p length with q difference between the diameter of y and p (or friction coefficient of r) and a initial source of air of s volume and a conection of t diameter (or a valve opening in u seconds)?" OR, how many psi do I need for a paintball gun?

I get it, but experimental measurements may be easier. If you calculated the force on the projectile over the distance it would be easy, But you'd have to make several assumptions about the actual pressure, friction, bypass, resistance from external air pressure etc.

F=MA or A=F/M

As an example 50 PSI on a 1" surface gives a Force of 50Lb. If the projectile is 1/16Lb your Acceleration is 50*16 (Diameter of pipe will give area through pi*r2)
Final velocity = root(2A*length of pipe travelled)

L

Ah, force. It's what pushes the projectile, it can be measured in pounds force, where it's equivalent to the weight pushing vertically downwards. I'm used to SI units where force is measured in Newtons (N).

Acceleration is the rate at which speed changes, i.e. how fast it goes from zero to final muzzle velocity. SI is meters per second per second (ms-2)

M is Mass (SI Kg)

The last is an equation of motion

L

The air pressure (PSI) is pushing the projectile, as PSI is Pounds per Square Inch, the amount of push (Force) depends upon how many square inches the compressed air is pushing against.

If you think of your tube, and a simple projectile (like a wine-cork) the air pushes on the end of the cork, in this case a circle about 3/4 of an inch across. The area of the circle (several burning questions Instructables on this) is given by pi x r2 which as an approximation is 3.1416 x (3/8 x 3/8) is about 0.44 square inches. "r" is the radius which is half the diameter (3/4 inch)

Multiply the PSI by the area calculated above and you have the force in pounds acting upon the projectile.

The force will cause the projectile to Accelerate, which can be quantified as the Force you calculate above divided by the mass of the projectile. More force = more acceleration, more mass means less.

The equation of motion is derived elsewhere (previous link), the speed at which the projectile leaves the tube is given by the square root (don't have symbol) of (2 x Accelleration x the length of pipe that the projectile travels)

Condensed:

Speed = sq root ( (2 x length of pipe x PSI x pi x r2) / mass)

(I think)

I'm probably not going to be able to do a better job than that, it's a bit tricky. But as I said earlier, experimental measurements may be better in real terms.

L