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

 

[Born2bwire]

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.

 

[rcgldr]

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).

 

[Andrew Mason]

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

 

[PJC01]

I would first like to commend you for taking the time to research and think about this problem. It shows a great dedication to understanding the mechanics involved.

To answer your question, the motion of a bullet inside a barrel can be described using the principles of Newtonian mechanics. The bullet will experience a net force due to the pressure difference between the gas behind it and the atmospheric pressure in front of it. This force will cause the bullet to accelerate forward, until it reaches the end of the barrel or is acted upon by another force (such as air resistance).

To find the equations of motion for the bullet, we can use Newton's second law, which states that the net force on an object is equal to its mass multiplied by its acceleration. In this case, the net force is the pressure difference between the gas and the atmosphere, and the acceleration is the change in velocity over time.

Using this information, we can set up the following equations:

F = ma
Pgas - Patm = m(dv/dt)

Where Pgas is the pressure of the gas behind the bullet, Patm is the atmospheric pressure, m is the mass of the bullet, and dv/dt is the change in velocity over time.

To solve for the equations of motion, we would need to know the initial pressure and volume of the gas behind the bullet, as well as the mass and shape of the bullet. We would also need to take into account any changes in the gas pressure as the bullet travels down the barrel.

In summary, the motion of a bullet inside a barrel can be described using the principles of Newtonian mechanics, taking into account the forces acting on the bullet and any changes in gas pressure. I hope this helps in your understanding of this problem.