Predicting force and velocity of a pneumatic projectile

[Crucible Guardian]

Here's a problem I've been working on for a while, and have made some breakthroughs on, but have recently gotten stuck. It stemmed out of my interest in pneumatic cannons. Consider a pneumatic cannon with a known inner surface area, a known barrel lenght, and with a projectile with a known mass. This cannon is pressurized to a known pressure. How can you predict the velocity of the projectile at any given moment?
Here's what I've got so far:

let P equal the original pressure.
Since pressure is equal to a force over an area, the force (F) acting on any given area is P*A.

The inside surface area of the pneumatic cannon changes as a function of x, which is the distance traveled down the barrel by the projectile. This fuction will be called A(x)

A(x) is equal to the original chamber surface area plus the amount of surface area 'revealed' by the distance x down the barrel by the projectile. The barrel is assumed to be cylindrical.

Let (Ao) equal the original chamber area
Let (Rb) equal the radius of the barrel

A(x)=(Ao+2*pi*Rb*x)

Therefore

F=P*(Ao+2*pi*Rb*x)

Since force is equal to mass times acceleration, acceleration is equal to force over mass

Let M equal the mass of the projectile
Let a equal the acceleration acting on the projectile

(P*(Ao+2*pi*Rb*x))/M=a
Acceleration is equal to the second-order derivative of x (the change in the change in distance with respect to time with respect to time)
so

Let t equal time

d2 x/d2 t =(P*(Ao+2*pi*Rb*x))/M
Since this will have to be integrated with respect to time, all x-values must be on the side of the derivative

multiply by M and divide by P
(d2 x/d2 t)*(M/P)=Ao+2*pi*Rb*x

subtract 2*pi*Rb*x and take each side to the e power. Using exponent rules, the left side can be broken up into a product of two exponents
(e^((d2 x/d2 t)*(M/P)))*(e^(-2*pi*Rb*x))=e^(Ao)

Does your head hurt yet?

Now the problem is, now that I've gotten this far, I have no idea how to integrate this. Any help would be greatly appreciated. I'm considering doing an analysis and test of this equation for a science fair, but I don't have to even register for it until late March, so it isn't imperative that this is solved immediatly.

 

[lippyka]

I haven't taken calculus yet, but I'm taking it next year so I would like to solve this problem so that it is fresh in my memory when I do take the class. To solve this problem, you need to use a numerical integration technique. The most commonly used is the Euler Method, which will help you approximate the velocity of the projectile at each time step. To do this, you first need to break down the equation into a set of first order ordinary differential equations. This can be done by dividing both sides of the equation by e^(-2*pi*Rb*x), and then taking the natural logarithm of both sides. This will give you an equation in the form of dy/dx=f(x,y). You can then use the Euler method to iterate through the equation and estimate the velocity of the projectile at each moment.

 

[quicknote]

I understand your frustration and commend you for your efforts so far. Your approach seems sound and I can see that you have a good understanding of the principles involved. However, I would suggest that instead of trying to integrate the equation, you may want to look into using numerical methods to solve it. This involves breaking up the problem into smaller steps and using a computer program or calculator to solve it. There are many resources available online for learning about numerical methods and how to apply them to physics problems. Additionally, you may want to consult with other scientists or engineers who have experience with pneumatic cannons and projectile motion to see if they can offer any insights or suggestions. Keep up the good work and don't be discouraged - sometimes the most challenging problems lead to the most exciting breakthroughs.