# Motion of a bullet inside a barrel

elegysix
I've spent the last hour or so googling this, and couldn't find anything straightforward. I thumbed through my thermo book and didn't come up with anything either. So I figured I'd ask you guys. I think this is a simple mechanics problem. I had been thinking about this last night, and figured it should be simple enough - I'm just getting stuck on the math somewhere I think.

Problem: Suppose there is a bullet of a given mass in a barrel at some initial position, with some initial volume and pressure of an ideal gas enclosed behind it. Then at some instant the bullet is allowed to accelerate due to the difference in pressures across the bullet. How do you find the equations of motion of the bullet ~ x(t), v(t), a(t) ? (x for position)

Assumptions: no friction between the bullet and barrel, and the bullet makes a perfect seal with the barrel. The temperature of the barrel is constant. The amount of gas is constant. Drag forces are negligible.

I can post my attempted solution if you guys want me too.

Thanks,
Austin

Gold Member
PV = nrT.

If the amount of gas is constant, and assuming temperature is constant, then the pressure should be inversely proportional to the volume of space behind the bullet. So, as the bullet travels down the barrel, the pressure should drop off. My guess is that you should assume that the bullet has some radius "a" and work out the amount of force exerted on the bullet as it travels the length of the barrel. From this you can easily work out the appropriate results.

Homework Helper
Even in the case of an ideal gas, as that gas expands, it's temperature will decrease, reducing the pressure at a greater rate than the rate of expansion. In the real world, the math is based on the results of previous measurments made with real guns and bullets (do a web search for internal and external ballistics).

Homework Helper
I've spent the last hour or so googling this, and couldn't find anything straightforward. I thumbed through my thermo book and didn't come up with anything either. So I figured I'd ask you guys. I think this is a simple mechanics problem. I had been thinking about this last night, and figured it should be simple enough - I'm just getting stuck on the math somewhere I think.

Problem: Suppose there is a bullet of a given mass in a barrel at some initial position, with some initial volume and pressure of an ideal gas enclosed behind it. Then at some instant the bullet is allowed to accelerate due to the difference in pressures across the bullet. How do you find the equations of motion of the bullet ~ x(t), v(t), a(t) ? (x for position)

Assumptions: no friction between the bullet and barrel, and the bullet makes a perfect seal with the barrel. The temperature of the barrel is constant. The amount of gas is constant. Drag forces are negligible.

I can post my attempted solution if you guys want me too.

Thanks,
Austin
The chemical reaction is very fast so the heat is released before the bullet has barely begun to move. What follows is essentially an adiabatic expansion (since it happens so quickly, there is very little heat flow from the gas).

The adiabatic condtion applies here: $$PV^\gamma = K$$ where $\gamma = C_p/C_v$. The pressure x cross-sectional area of the barrel = the force on the bullet = mass x acceleration. So:

$$\ddot{x} = PA/m$$

See if you can work out the rest from that.

AM