Thermodynamics isothermal expansion problem that wants us to find initial volume

[eggheadd]

Homework Statement

A projectile of mass M kg is accelerated from rest to V m/s using a compressed air cannon. conceptually, we may consider the projectile to be a frictionless "piston" within a cylinder that is closed at one end and open to the atmosphere at the other end.

Before firing, the projectile is pushed into the cylinder where it is secured by some sort of catch. during this process a mass, m, of air at a pressure of P kPa(absolute) is trapped behind the "piston". The catch holding the "piston" is released, allowing the air to push it down the barrel of the "gun" until it emerges at its final velocity.

Assume that the expansion is isothermal. Find the miniumum volume V1(L) needed to provide a velocity 42 m/s at the gun exit, given that the initial pressure P1 is 899 kPa, atmospheric pressure P0 is 100 kPa and the "piston" mass M is 10 kg.

The answer for this question is 7.504, it already got graded, I'm just trying to figure out how it's solved. Does anyone know how?

Homework Equations

Cons of energy: Q-Wother= ΔU +ΔKE +ΔPE
Work= P1V1ln(V2/V1)
ΔKE= .5(M)v^2

The Attempt at a Solution

I know that ΔPE is 0 and Q is 0 since there's no heat put into the system. so i end up with

-P1V1ln(V2/V1)=mgas(u2-u1) +.5(M)v^2
but i don't know V1,V2, or mgas. Can anyone help me with this. The answer is 7.504 but i don't know how the prof got that answer?

 

[rude man]

1. Why do you assume Q=0? It's an isothermal process, not an adiabatic one.

2. What is the temperature T?

3. Next step is to compute p₂, the pressure difference between the mass inside the barrel and atmospheric just before exiting the barrel. That is readily done.

3. BUT - I think (right now) that this problem is undefined. One is left with 3 unknowns (V1, V2 and m) and only two state equations. If anyone of those three is known, the rest is straightforward. Maybe I'll think of something else later. Did you state the problem verbatim as given to you? I'm suspicious since you didn't mention T, for example.