Pressure in an air cannon- ODE

[J Hill]

Pressure in an air cannon-- ODE

Homework Statement

Okay, I'm designing an air cannon, and need to calculate the muzzle velocity. The only propelling force is the one caused by the pressure of the compressed air.

Homework Equations

I've been working with Boyle's law:
$$P_0 V_0 = P_1 V_1$$

The Attempt at a Solution

From here, I can express pressure as a function of distance down the barrel
$$P_1 = \frac{P_0 V_0}{V_1}$$
$$P_1 = \frac{P_0 (\pi r^2 x_0)}{\pi r^2 x}$$
here x represents the initial length of of the chamber inside the barrel where the pressure builds up plus the distance down the barrel.
From this, I get:
$$P = \frac{P_0 x_0}{x}$$
And working with the definition of pressure and Newton's second law:
$$\frac{m a}{\pi r^2}=\frac{P_0 x_0}{x}$$
$$a = \frac{P_0 x_0 \pi r^2}{m x}$$
This is the part I can't get passed:
$$x'' = = \frac{P_0 x_0 \pi r^2}{m x}$$
I haven't yet taken a course on differential equations, so I don't know how solve this: all of the values, except x, are constant, so there has to be a general solution to
$$x'' = \frac{1}{x}$$
Can anyone tell me what it is?

 

[mighty2000]

Hello there! As a fellow scientist, I can offer some guidance on your problem. First, I would like to commend you on your use of Boyle's law to calculate the pressure in the air cannon. However, I believe you may have overlooked a key factor in your calculation.

The pressure in the air cannon is not constant as the air is being released and propelling the projectile. As the air is released, the volume of the chamber decreases, causing an increase in pressure. This means that your equation for pressure should also include the change in volume over time (dV/dt).

Additionally, you will need to consider the mass of the projectile and the force acting on it (F=ma) in order to calculate the muzzle velocity. This will require some manipulation of your equations to solve for the variables of interest.

I would recommend taking a look at the ideal gas law (PV=nRT) and the equation for work (W=Fd) to help you in your calculations. And don't worry about the differential equation for now, as it may not be necessary for this problem.

I hope this helps and good luck with your air cannon design!