How Does Firing Cannonballs Inside a Railroad Car Affect Its Position and Speed?

[Coldie]

Hey guys,

My first question is as follows:
A cannon and a supply of cannonballs are inside a sealed railroad car of length L. The cannonballs remain in the car after hitting the far wall.
a) After all the cannonballs have been fired, what is the greatest distance the car can have moved from its original position?
b) What is the speed of the car after all the cannonballs have been fired?

Since there are no values provided, this seems to be a thought exercise. While the cannonballs are in the air, the railroad car is moving in the opposite direction. When the cannonballs hit the other side of the car, however, they will probably bounce off, causing the car to move back in the other direction, and the car will stop when the cannonballs again rest on the floor of the car. With ideal values, I'm assuming the car end up right back where it was originally, but I'm not sure how to verify this. The conclusion that both the displacement and speed of the car after firing the cannonballs is zero follows from my conclusion. Are my impressions correct?

Next question:
Find the center of mass of a homogeneous semicircular plate. Let R be the radius of the circle.

There's an example earlier in the book that shows how to find the center of mass of a thin strip of material bent into the shape of a semicircle. I'm wondering if the center of mass of that shape is the same as the one of a semicircular plate. If so, then I can simply use the example in the book for the answer. Is the center of mass the same in both cases?

Help on either of these questions would be appreciated!

 

[Doc Al]

My first question is as follows:
A cannon and a supply of cannonballs are inside a sealed railroad car of length L. The cannonballs remain in the car after hitting the far wall.
a) After all the cannonballs have been fired, what is the greatest distance the car can have moved from its original position?
b) What is the speed of the car after all the cannonballs have been fired?

Since there are no values provided, this seems to be a thought exercise. While the cannonballs are in the air, the railroad car is moving in the opposite direction. When the cannonballs hit the other side of the car, however, they will probably bounce off, causing the car to move back in the other direction, and the car will stop when the cannonballs again rest on the floor of the car. With ideal values, I'm assuming the car end up right back where it was originally, but I'm not sure how to verify this. The conclusion that both the displacement and speed of the car after firing the cannonballs is zero follows from my conclusion. Are my impressions correct?

Since no external forces act on the car, you can say that the total momentum of the car (and its contents) must be conserved. You can conclude that the position of the center of mass cannot have moved, but that doesn't mean that the car hasn't moved.

Take another example: You are standing on one end of a rowboat floating in the water. If you walk to the other end of the boat, will the boat move? It's a similar situation. While the center of mass of "you + boat" doesn't change (ignoring any resistance of the water), in order to move forward you and the boat essentially push each other away. (Of course, you stop each other before you walk off the boat.)

The same thing happens with the cannonballs and the car. When the balls are shot from the cannon, the car moves back. When the balls smack into the wall, the motion stops, but the balls (presumably) are now on the other end of the car. So, does the position of the center of mass (of everything) measured with respect to the car change? (Hint: The initial and final configurations of cars and balls are mirror opposites of each other.)

Next question:
Find the center of mass of a homogeneous semicircular plate. Let R be the radius of the circle.

There's an example earlier in the book that shows how to find the center of mass of a thin strip of material bent into the shape of a semicircle. I'm wondering if the center of mass of that shape is the same as the one of a semicircular plate. If so, then I can simply use the example in the book for the answer. Is the center of mass the same in both cases?

A semicircular plate can be thought of as a collection of thin semicircular strips, but each will strip will have a different radius and center of mass. You'd have to integrate to find the center of mass of the entire plate.

 

[Coldie]

Ah, the center of mass does not change, but the position of the car may! I understand now. Should I say that the car would move proportionally to the amount of mass that is displaced within the system by the fired cannonballs, or is there a better response?

The equation we're given in the book for finding the center of mass of a semicircle is $$\frac{1}{M}\int_{0}^{\pi}(R\sin\phi)\frac{M}{\pi}d\phi$$

The answer they obtained in the book was .637R. Is this also true of a semicircular plate?

Thanks very much for the response!

 

[Doc Al]

Ah, the center of mass does not change, but the position of the car may! I understand now. Should I say that the car would move proportionally to the amount of mass that is displaced within the system by the fired cannonballs, or is there a better response?

You can make the answer as precise as you wish by making some simplifying assumptions: Call the length of the car L, its mass Mc, the mass of the balls Mb. Assume the COM of the car itself is at its midpoint. Now you can calculate the maximum shift of the car due to the balls moving from one side to the other. (Of course, external forces--like friction--may prevent the car from sliding on the tracks. That's why they speak of "greatest distance".) What if Mc >> Mb? Or the reverse: Mb >> Mc?

The equation we're given in the book for finding the center of mass of a semicircle is $$\frac{1}{M}\int_{0}^{\pi}(R\sin\phi)\frac{M}{\pi}d\phi$$

The answer they obtained in the book was .637R. Is this also true of a semicircular plate?

No. That describes a thin semicircular strip, not a plate.

 

[Coldie]

Thanks, my only problem now remains the second part. I'm having great difficulty coming up with an equation for the circle, which seems absurd. The equation for center of mass in the y-axis is in the form$$\frac{1}{M}\int_{a}^{b}$$

I'm not sure what y, nor dm, are to be. Could you please help?

Thanks!

 

[Doc Al]

You can start with that expression for the center of mass of semicircle and modify it to be a double integral, since you want to integrate over all radii from 0 to R:
$$\frac{1}{M}\int_{0}^{\pi} \int_{0}^{R}(r^2\sin\phi)\frac{2M}{\pi R^2}d \phi dr$$

Where $$dm = \frac{2M}{\pi R^2} r d\phi \dr$$

(But you'd better check it!)

 

[zhangman960]

Here's a solution to the center of mass of a semicircular plate using Cartesian coordinates:

http://www.cramlab.com/Resources/Math_Problems/SemicircleCentroid.html

Hope that helps!