Recently, I made a homemade air powered airsoft gun. I wanted to know how fast it would accelerate out of the barrel, so I tried do do some calculations, but to no avail (keep in mind, I haven't taken physics). I started with a 0.2g and 6mm bb and 75psi of air pressure. First I converted the air pressure to pascals, which turns out to be about 500,000 Pa. Since a pascal is equivalent to one newton per meter squared, I rewrote it as this: $\frac{500,000(kg)*m}{s^2*m}$. I calculated that the air pressure would be exerted over the back half of the bb, which turns out to be about $56mm^2$ which equals $5.6*10^{-5} m$. After multiplying the 500,000 pascals by the surface area, my answer results in: $\frac{28(kg)*m}{s^2}$. Next I intuitively divided the mass of the bb, which is 0.0002 kg, in order to get rid of the mass unit. (I then realized that I was using f=ma to solve for a) After that calculation, my answer was $\frac{140,000m}{s^2}$. Clearly, even void of friction, this answer cannot be correct since my answer would suggest that after it exited the barrel, it would be traveling over mach 1000. Can someone point me in the right direction to find the correct answer?
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2$\begingroup$ The area of a round 6mm bb comes out to about $\pi r^2\approx 3*3*3.14 mm^2 \approx 28mm^2$. How did you get the $250mm^2$? That's 14N of force and an initial acceleration of 7000g. For a 50cm long barrel that gives me 132m/s as velocity. Sounds about right, no? It couldn't go much faster than that, anyway, otherwise one would have to understand sonic and even hypersonic airflow in the gun and that's not so easy (and not happening here). $\endgroup$– CuriousOneCommented Jan 11, 2016 at 22:32
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$\begingroup$ Ah good call. I used 6 as the radius instead of three. But your surface area equation is for a circle not a sphere. I will update my question :) @CuriousOne $\endgroup$– RyanCommented Jan 12, 2016 at 0:02
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2$\begingroup$ Ah... did you use the area of a half sphere? OK. While tempting it's only the circular cross section that counts. The pressure does act on the normals of the sphere, but the components of that force that are not pointing in the direction of the barrel do not accelerate the bb. All of this is minor, though. How did you calculate the total velocity? I think that's where the major problem in your estimate is. $\endgroup$– CuriousOneCommented Jan 12, 2016 at 0:09
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$\begingroup$ I calculated the acceleration actually, since my units are in m/s^{-2}. With my force of 28N and mass of .0002kg (.2g), I basically used f=ma to solve for a. @CuriousOne $\endgroup$– RyanCommented Jan 12, 2016 at 0:17
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1$\begingroup$ The acceleration is very high... but the barrel is short. If you use $s=at^2/2$ and $v=at$, then you can substitute $t=v/a$ in the first equation, which gets you $s=a(v^2/a^2)/2=v^2/2a$. It follows that $v=\sqrt{2as}$, which is dimensionally correct, if you check the units. If we insert an acceleration of $a=70,000m/s^2$ and a barrel length of $s=0.5m$, then we get $v=\sqrt{2*70000m/s^2*0.5m}=265m/s. And now I have to apologize because I got it wrong by a factor of 2 the first time (I had it in the denominator instead of the numerator under the square root). $\endgroup$– CuriousOneCommented Jan 12, 2016 at 0:36
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