How Do You Calculate the Initial Pressure in an Air Rifle Using Thermodynamics?

[thenewbosco]

an air rifle shoots a lead pellet by allowing high pressure gas to expand.
This happens quickly so it is an adiabatic process. initially 12 cubic centimeters of air are admitted to the barrel. It behaves as an ideal gas, $$\gamma = 1.40$$. The air expands behind a 1.1 g pellet and pushes on it as a piston with cross sectional area 0.03 cm^2. the pellet emerges from the barrel with speed 120 m/s.
I am to use the work gone by the gas:
$$W = (\frac{1}{\gamma - 1})(PiVi - PfVf)$$
to calculate the initial pressure required.
I do not know though how to find the final volume and final pressure...

I thought to use $$PfVf^\gamma = PiVi^\gamma$$ but this does not seem to change anything for me i still need to know Vf...

any help on this...thanks

 

[maverick280857]

Can you relate the time rate of change of momentum of the pellet to the average force exerted on it? And who would exert it (rather what)?

Also while deriving the equation for work done in an adiabatic process, you have already taken into account that $P_{i}V_{i}^{\gamma} = P_{f}V_{f}^{\gamma}$.

Hold on a sec: you know the mass of the pellet, its initial and final speeds so you can calculate the the change in momentum. To calculate the force, you need to find the rate of change of momentum. You need either the acceleration of the pellet or the time it took to attain a speed of 120 m/s. I think something is missing here. But if I get some more ideas, I'll post them here soon.

 

[Gokul43201]

Bosco, you have 2 equations in 2 unknowns. What's the problem ?

 

[GCT]

what is the distance for which the bullet was accelerated?

 

[thenewbosco]

oh i forgot the distance is 50cm.

 

[GCT]

so what's the change in volume?

 

[thenewbosco]

i said that it was 12cm^3 (the initial volume admitted) + (0.03cm^2)(50)cm.

I would then put this into the Vf for final volume but i then need the final pressure which i don't know...

 

[thenewbosco]

anybody?

 

[GCT]

just use substitution, right?

 

[quantumdude]

bosco,

I've merged your two threads together. Don't post the exact same thread multiple times. Once is enough.