the final velocity non-constant acceleration

Question

I'm not sure if my problem was asked before, but if so, a simple link to the answer would be appreciated.

Basically I've recently seen videos about spud cannons that use compressed air and it got me thinking about the math behind it.

Obviously for targeting purposes, one needs the final velocity at the end of the cannon barrel to substitute in newton's projectile motion equations and get the distance which is cool.

Now to find the velocity at that point is where it gets really messy. Since the pressure of compressed air changes as it expands throughout the cannon barrel, the force it exerts on the projectile decreases and so does the acceleration.

With the assumption of an isothermal expansion of the air in the barrel, how do you find the end velocity of the projectile at the end of the cannon barrel?

Answer 1

Well, isothermal is a bad assumption for an ideal gas in this situation since the energy in the gas is $\frac{3}{2}k_bT$ but we know the energy should change since the gas propels an object.

But it is the simplest assumption we can make so lets go from there anyways.

$$F=PA=ANk_bT/V=\frac{ANk_bT}{Ax}=\frac{Nk_bT}{x}$$

Now we just integrate this to get the work which is the final kinetic energy:

$$W=Nk_bT\ln\left(\frac{x_f}{x_i}\right)=1/2mv_f^2$$

$$\sqrt{\frac{2}{m}Nk_bT\ln\left(\frac{x_f}{x_i}\right)}=v_f$$

Now if you wanted to consider change in temperature as you expand, you need to write $T$ as $$T_i-\Delta T=T_i-\frac{2}{3k_b}|K_{gas_i}-K_{gas_f}|=T_i-\frac{2}{3k_b}W$$

Then you can take a derivative of the force and get a differential equation for $\frac{df}{dx}$. Now you solve this equation for force, get acceleration and integrate for final velocity.