Cannon Ball Chain Shot Impulse Problem

[sirzerp]

Homework Statement

Image you have one of the old fashion chain shots from the 1800’s where they had two cannon balls connected by a chain. Now let’s say you wish to try to fire the chain shot from two cannons. The synchronized cannon firing nearly works, but one cannon ball receives a different impulse acceleration than the other cannon ball. Assume the chain is always tight and the cannons balls are always a constant distance away from each other.

The starting velocity is zero, they are at rest. Assume we know the mass and starting impulse accelerations exactly. Assume the barrel length does not effect the system, that the cannon balls are free to move in any direction after the starting impulse as long as they stay one unit away (fixed body) from each other.

Given the rest locations of p1,p2 and impulse accelerations a1,a2, and mass of m1,m2.

At what locations is p1 and p2 after time t?

Homework Equations

F=mA

The Attempt at a Solution

I am writing a computer program and need to program the general solution to the above problem. It is a self posed problem and not for class credit. Trying to avoid spending a day or two with Goldstein.

Anyone know the general solution to this dual mass impulse problem?

 

[mark726]

Hello there! I am always interested in exploring new problems and finding solutions. In this case, the problem you have presented is quite interesting and challenging. After some thought and calculations, I believe I have found a general solution to the dual mass impulse problem you have described.

Firstly, we need to understand the concept of impulse and its relation to momentum. Impulse is defined as the change in momentum of an object, and it is equal to the force applied multiplied by the time it is applied for. In this case, we have two cannon balls connected by a chain, so the impulse applied to one ball will also be applied to the other, but in opposite directions.

Let's denote the impulse applied to the first cannon ball as I1 and the impulse applied to the second cannon ball as I2. Since the cannon balls are connected by a chain, the total impulse applied to the system will be the sum of these two impulses, which we can write as I = I1 + I2.

Now, using the equation F = mA, we can express the impulse applied to each cannon ball in terms of their mass and acceleration. For the first cannon ball, we have I1 = m1 * a1, and for the second cannon ball, we have I2 = m2 * a2.

Since we know the starting impulse accelerations and the mass of each cannon ball, we can calculate the total impulse applied to the system. This will give us the total change in momentum of the system, which we can then use to determine the final velocities of each cannon ball.

To do this, we can use the equation p = mv, where p is the momentum, m is the mass, and v is the velocity. Since the cannon balls are initially at rest, their initial momentum will be zero. Therefore, the final momentum of the system will be equal to the total impulse applied, which we calculated earlier.

Once we have the final momentum, we can use it to calculate the final velocities of each cannon ball. This can be done using the equation p = mv, where p is the final momentum, m is the mass, and v is the final velocity. We can solve this equation for v and substitute in the values for the final momentum and the mass of each cannon ball.

The final step is to determine the positions of each cannon ball after a certain amount of time, t. This can be done using the equation x = vt, where x