An airgun shoots a ball made from lead, through air of high pressure that pushes the ballforward like a piston. Because this happens very quickly, Q = 0 and the process is adiabatic. The initial volume of the air is Vi = 12.0 cm3 = 12.0 * 10-6 m3, which behaves as an ideal gas with γ = 1.40. The air decompresses and pushes the ball as a piston through the gun's barrel, which has a length L = 50.0 cm = 0.500 m. The ball has a cross section area A = 0.0300 cm2 = 0.0300 * 10-6 m2and has a mass m = 1.10 g = 1.10 * 10-3 kg. Which is the initial pressure Pi so that the ball can exit the barrel with v = 120 m/s ? Ignore any friction from the barrel or the win'ds resistance.
PiViγ = PfVfγ
∑F = ma
P = F/A
V=4/3 * π‧r3 (ball)
Lead's Density: p = 11.34 g/cm3 = 11.34 * 103 kg/m3
The Attempt at a Solution
Alright, my first instict was that I'd use this formula: PiViγ = PfVfγ
Now, to use this, I need to find Vf & Pf, so that coupled with Vi & γ (which I already know), I can find Pi.
So, I can use L & A to find Vf, like this:
>p = m/V <=> 11.34 * 103 kg/m3 = 1.10 * 10-3 kg / V <=> V = 9.7 * 10-8 m3 (the ball's volume)
>Ball: V=4/3 * π‧r3 <=> ... <=> r = 2.85 * 10-3 m
>Cylinder: V=π‧r2‧L = 1.28 * 10-5 m3
So Vf = 1.28 * 10-5 m3
The problem I'm facing is that I can't figure out how to connect Pf with the v (velocity).
Any help is appreciated!
PS: Could somebody check the calculations a bit? If I take the V = A*h = A* L formula, I get V = 1.5 * 10-6 m3, which is the volume of the whole barrel, minus the ball. Now, I get Vf = 1.28 * 10-5 m3 which is the volume of barrel with the ball blocking a part of it. So, logically, the volume of the ball should be Vb = 22.0 * 10-6 m3, but as we can see above, using the density and mass I get Vb = V = 9.7 * 10-8 m3.
Can somebody spot the mistake before I move on and do a mess at the folowing calculations?