[Help] creating the biggest force with air pressure?

[kieyard]

hello, I'm wanting to make an air pressure cannon and am trying to mathematically find out how to get the highest muzzle velocity, could someone please check over my working and let me know of any mistakes and perhaps offer suggestions on how to improve my cannon.

if my maths is correct am i also right in saying that to get the maximum force out of it R and r should both be below 1?

any guidance would be appreciated.

 

[kieyard]

i also done some excel work to calculate the final velocity. i did some preliminary tests before this to find out the best optimum values for R, r and L. could you also check if the working is correct on these? iv included the formula view, so you can check.

 

[Khashishi]

It looks like you are using the wrong area. You want to use the area of the base of the bullet = pi*r^2, if you are calculating the force on the bullet.

Also, the equation for pressure is missing temperature. Temperature will decrease as volume increases. You might be able to approximate it as an adiabatic expansion, in which case your equation should be
$P = \frac{c}{V^\gamma}$
$\gamma$ depends on what gas you are using in your chamber. $\gamma = 5/3$ for a monatomic gas like helium or $7/5$ for a diatomic gas like air.

You should always take a step back and think about what you expect the answer to be, and if your answer makes sense at all.
This makes no sense: "if my maths is correct am i also right in saying that to get the maximum force out of it R and r should both be below 1?"

 

[kieyard]

yeah, i realized the area thing the day after writing this and to get a greater force the volume should be the smallest as possible hence why i said the R and r below 1 because it grows exponentially if its above 1 however the smaller the volume the less mols i can fit into it and when the projectile is at the highest the volume is massive compared to its original meaning the final force will be small.
plus the temperature will only increase as pressure increase which would also increase the force and the idea is to get the highest force so even though i missed this its not too much a problem. it would be nice to include it in my maths however.

 

[Khashishi]

It made my brain hurt to read your first "sentence". Please use punctuation and capitalization. Even when I add punctuation it doesn't really make any sense. Your physics intuition is wrong. First of all, when you are working with real units with real dimensions, the number 1 doesn't mean anything special. For length, you need a unit, like 1 meter. And it sounds like you just pulled the 1 out of nowhere. Why 1 and not 2 or 0.483? There is no exponential growth in your equation. Exponential growth means $e^x$.

For a given tank pressure, a larger volume tank will give you more impulse than a smaller tank because the pressure will not decrease as quickly as the bullet moves through the gun barrel.