# Forces in a paintball gun

## Main Question or Discussion Point

How do you calculate the forces experenced upon a paintball when being fired through the barrel of a paintball gun?
Barrel lenght being 12"
final velocity being 100m/s^2

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$$F=ma=m\frac{dv}{dt}$$
You would have to know the time in which the force that causes acceleration is working and the mass of the paintball.

PS. The unit of velocity is m/s, not m/s²

How do you calculate the forces experenced upon a paintball when being fired through the barrel of a paintball gun?
Barrel lenght being 12"
final velocity being 100m/s^2
You would need to know the mass of the paint ball. You can also get its kinetic energy using the formula for kinetic energy.

$$KE=\frac{1}{2}$$$$mv^{2}$$

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You would need to know the mass of the paint ball. You can also get its kinetic energy using the formula for kinetic energy.

$$KE=\frac{1}{2}$$$$mv^{2}$$
And KE = Work = Fs
So then you would need to know the distance on which the force works. So you either need to know the time interval or the distance. I assume that the bullet isn't being accelerated througout the entire barrel, but I'm not a weapon expert.

And KE = Work = Fs
So then you would need to know the distance on which the force works. So you either need to know the time interval or the distance. I assume that the bullet isn't being accelerated througout the entire barrel, but I'm not a weapon expert.
Why would you need the time interval? The paint ball is not being accelerated. It is struck with burst of CO2 in the beginning.

After that the paint ball is just moving on the energy it received from the initial burst.

russ_watters
Mentor
You guys are going around in circles a bit and the approaches both work (though the second, not quite as described): if you know the force and the distance (we were given the distance, we'd need the CO2 pressure and how/if it varies as the ball moves through the barrel), you easily calculate the work and since work is equal to kinetic energy, you can use the kinetic energy equation to calculate the final speed. In my opinion, that is the most direct method.

Alternately, you can use newton's mechanics equations, a=f/m, s=at, and d=st, with some integration, to find the same thing. You don't know the time: you have to calculate it.

Dale
Mentor
In any case, you need the mass of the paintball.

I would use the KE approach and just assume that it is accelerated by a constant force down the entire length of the barrel. That will surely underestimate the true maximum force, but should be a decent first-order approximation.

If you want a more advanced approximation then you can consider the CO2 to be an adiabatically expanding ideal gas and assume that no CO2 leaks around the ball in the barrel. That should give you a force that varies down the barrel length in a reasonable manner.

If you want a more advanced approximation then you can consider the CO2 to be an adiabatically expanding ideal gas and assume that no CO2 leaks around the ball in the barrel. That should give you a force that varies down the barrel length in a reasonable manner.
How would you set up an equation that takes care of that?

Dale
Mentor
Here is the Hyperphysics page with the equation for an http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/adiab.html#c1".

You know the total work done on the ball so start with an initial guess for the starting volume, add the volume of the barrel to get the final volume, and use that to solve for K. Then you can use K to solve for the pressure as a function of the volume, which will give you the force on the paintball as a function of distance down the barrel.

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