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I have an air powered rifle and I would like to find a way to calculate the amount of force it produces as well as its velocity. The air rifle can be pumped which increases the pressure which gives the metal bb a lot more energy. I dont think there is any real way I could reliably test for acceleration but it might be possible to find its velocity using a 240 fps camera and enough distance but it will still be tricky.

Another idea I had was to calculate a distance then use a laser level to find where the projectile will hit a target then calculate how much the bb dropped. For example I could use $H = V_0 t + \frac 12 a t^2$ then plugin $0$ for $V_0$ which would remove the first $t$ and get me $H = \frac12 a t^2$ then put in the height for $H$ and gravity for $a$ to find the time it took to reach the target. Then I could take the distance traveled with the time to find its velocity (which assumes its constant even though its not due to minimal wind resistance).

That being said what is the best way to find its force? I have no idea how to find its acceleration but could I just use the barrel length with the velocity I calculated earlier to find its acceleration since it started from rest and then use that acceleration with its mass to find force?

If what I said above is correct how can I find the actual force, velocity and acceleration while taking into account wind resistance? I have taken multiple calculus courses but I am not exactly sure the best way to find this value.

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  • $\begingroup$ At these speeds you can't neglect air drag. And trust me: that really complicates the math! $\endgroup$
    – Gert
    Mar 14 at 21:39
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    $\begingroup$ You could shoot a target hanging on a string close to the exit of the barrel, like a lump of clay, and see how far it swings up. You could also measure the recoil of the gun, either with a real-time force sensor, or just hang the gun using a string at each end. The lump of clay will be easier. $\endgroup$
    – tom10
    Mar 14 at 21:39
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    $\begingroup$ @tom10 My thoughts entirely. Measurements... $\endgroup$
    – Gert
    Mar 14 at 21:40
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    $\begingroup$ @UnSure No, you use conservation of momentum to calculate muzzle velocity. $\endgroup$
    – Gert
    Mar 14 at 21:44
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    $\begingroup$ en.wikipedia.org/wiki/Ballistic_pendulum $\endgroup$
    – Gert
    Mar 14 at 21:48
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The quantity to best consider measuring is the momentum of the BB (and the velocity can then be found by dividing the momentum by the mass). An easy way to measure the momentum is using a ballistic pendulum. In short, hang a block of material that will capture the BB directly in front of the barrel of the gun (but far enough away so the air from the gun doesn't move the block). Then fire the BB and measure how high the block rises, and apply the formula in the link.

To optimize this you want a material that will capture the BB and not be so heavy that it's hard to measure the rise. It may require trying a few different materials so that the BB is captured and the block is in the correct weight range. (For example, clay might work as a capture medium, but you might find that using enough clay to capture the BB requires a slab that's so thick that it doesn't lift high enough to easily measure, so maybe then try wood.)

The question refers to "measuring the force of the BB", but the BB does not exactly carry force, it carries momentum. Momentum relates to force in that when the BB strikes something, it will slow down due to a resisting force applied by the material it strikes, and this force applied over the time that the BB is slowing down is related to the momentum carried by the BB, but the force itself is as dependent on the materials as it is on the speed of the BB. For example, if the BB strikes something hard, the force will be very large and short, or if it strikes something soft, the force will be small but applied over a long time.

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