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## Homework Statement

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 cm

^{3}= 12.0 * 10

^{-6}m

^{3}, 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 cm

^{2}= 0.0300 * 10

^{-6}m

^{2}and 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.

## Homework Equations

PiVi

^{γ}= PfVf

^{γ}

∑F = ma

P = F/A

p =m/V

V=π‧r

^{2}‧h (cylinder)

V=4/3 * π‧r

^{3}(ball)

Lead's Density: p = 11.34 g/cm

^{3}= 11.34 * 10

^{3}kg/m

^{3}

## The Attempt at a Solution

Alright, my first instict was that I'd use this formula: PiVi

^{γ}= PfVf

^{γ}

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 * 10

^{3}kg/m

^{3}= 1.10 * 10

^{-3}kg / V <=> V = 9.7 * 10

^{-8}m

^{3}(the ball's volume)

>Ball: V=4/3 * π‧r

^{3}<=> ... <=> r = 2.85 * 10

^{-3}m

>Cylinder: V=π‧r

^{2}‧L = 1.28 * 10

^{-5}m

^{3}

So Vf = 1.28 * 10

^{-5}m

^{3}

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}m

^{3}, which is the volume of the whole barrel, minus the ball. Now, I get Vf = 1.28 * 10

^{-5}m

^{3}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}m

^{3}, but as we can see above, using the density and mass I get Vb = V = 9.7 * 10

^{-8}m

^{3}.

Can somebody spot the mistake before I move on and do a mess at the folowing calculations?