- #1

RazerM

- 6

- 0

## Homework Statement

1.75 J is required to fire a spherical pellet of radius 8.75 mm at a maximum velocity of 100 m/s

Find appropriate values for the inital pressure and volume and final pressure and volume

## Homework Equations

Adiabatic Expansion

[tex]P_{1}V_{1}^{\gamma}=P_{2}V_{2}^{\gamma}[/tex]

[tex]\gamma=1.40\qquad P_{2}=10^5\text{ Pa}[/tex]

[tex]W_{\text{total}}=\frac{P_{1}V_{1}-P_{2}V_{2}}{\gamma-1}[/tex]

[tex]

\begin{align*}

W_{\text{total}} &= W_{\text{atm}}-W_{\text{useful}} \\ \frac{P_{1}V_{1}-P_{2}V_{2}}{\gamma-1}&=10^{5}(V_{2}-V_{1})-1.75

\end{align*}

[/tex]

## The Attempt at a Solution

[tex]

\begin{align*}

V_{2}&=V_{1}\left( \frac{P_{1}}{P_{2}} \right)^{\frac{1}{\gamma}}\\

V_{2}&=V_{1}\left( \frac{P_{1}}{10^{5}} \right)^{\frac{1}{1.4}}\\

\end{align*}

[/tex]

It is then possible to rearrange work equation into

[tex]V_{1}\left( [P_{1}+4\cdot 10^{4}]-\left[ 1.4\cdot 10^{5}\left( \frac{P_{1}}{10^{5}} \right)^{\frac{1}{1.4}} \right] \right)=0.7[/tex]

[tex]\text{Then using an arbitary guess for }V_{1}\text{ and an iterative process to find }P_{1}\text{ I found }[/tex]

[tex]

V_{1} = 10^{-5}\text{ m}^{3}[/tex]

[tex]

P_{1} = 420465\text{ Pa}[/tex]

[tex]

V_{2} = 2.79\cdot 10^{-5}\text{ m}^{3}

[/tex]

Giving a ridiculous barrel length of

[tex]

\begin{align*}

L &= \frac{V_{2}-V{1}}{\pi r^{2}} \\

L &= 0.0744\text{ m}

\end{align*}

[/tex]

I'm not asking for a full solution; just a hint as to how to find out values that satify the criteria of work and have a reasonably small V_1 so that there can be many shots for say a 0.001m^3 tank