A 500-kg cannon and a supply of 80 cannon balls, each with a mass of 45.0 kg, are inside a sealed railroad car with a mass of 11000 kg and a length of 49 m. The cannon fires to the right; the car recoils to the left. The cannon balls remain in the car after hitting the wall. After all the cannon balls have been fired, what is the greatest distance the car can have moved from its original position?
What is the speed of the car after all the cannon balls have come to rest on the right side?
The center of mass of car and contents remains in the same place.
For greatest amount of displacement, assume that all the cannonballs move from one end to the other
The final speed is zero, since momentum is conserved
I don't quite follow
Well, let's calculate how far the car has moved first. We can use the principle of conservation of momentum to find the distance.
The initial momentum of the system (cannon, cannonballs, and car) is zero since everything is at rest.
Let's assign positive velocities to the right and negative velocities to the left.
The momentum of the cannon is 0 kg m/s since it has no initial velocity.
The momentum of the car is given by its mass (11000 kg) multiplied by its final velocity, which we'll call Vc.
The momentum of the cannonballs is given by their mass (80 * 45 kg) multiplied by their final velocity, which we'll call Vb.
Since momentum is conserved, we have:
0 + (11000 kg * Vc) = (80 * 45 kg * Vb)
Simplifying, we find:
11000Vc = 3600Vb
Now, it's stated that the cannonballs come to rest on the right side, meaning their final velocity, Vb, is zero.
Therefore, we can solve for Vc:
11000Vc = 0
Vc = 0 m/s
So, the car doesn't move at all!
Now, to answer your second question, the final velocity of the car is 0 m/s since it doesn't move. Therefore, the speed of the car after all the cannonballs have come to rest on the right side is also 0 m/s.
I hope that answers your questions! If you need any more assistance, feel free to ask!
The problem wants you to assume that the railroad car in on a frictionless track. Such cars do not exist, but assume this one rolls without friction, anyway. I hope you have heard of the law of conservation of momentum.
The initial momentum of the car and its contents is zero. The total momentum of the car and its contents must remain zero. When nothing moves inside the car, the car must also have zero velocity.
The first part of your question is harder to explain. In a problem such as this where you are not dealing with a rigid body, the momentum conservation equation still applies if for total momentum you use the total mass times the velocity of the center of mass.
In your problem, the center of mass does not move in Earth-based coordinates, because of the momentum conservation law. The train will move temporarily while the shooting is going on, and stop when it is over. The center of mass as seen from inside will move, but from outside it will not.
The motion of the center of mass as seen from inside equals the final displacement of the car.
To find the greatest distance the car can have moved from its original position after all the cannon balls have been fired, we need to apply the principle of conservation of momentum.
First, let's calculate the initial momentum of the system, which includes the cannon, the cannonballs, and the railroad car. The momentum can be calculated using the formula:
Initial momentum = (mass of cannon + total mass of cannonballs + mass of car) × initial velocity
The initial velocity can be assumed to be zero because the system is at rest initially.
Initial momentum = (500 kg + 80 cannonballs × 45 kg each + 11,000 kg) × 0 = 0
Since the initial momentum of the system is zero, the final momentum must also be zero to satisfy the conservation of momentum principle. This means that the momentum of the car moving to the left must be equal in magnitude to the momentum of the cannonballs moving to the right.
The momentum of the cannonballs can be calculated using the formula:
Momentum of cannonballs = total mass of cannonballs × final velocity
Since all the cannonballs come to rest on the right side, the final velocity of the cannonballs is zero.
Momentum of cannonballs = 80 cannonballs × 45 kg each × 0 = 0
To maintain a final momentum of zero, the mass of the cannon & car system must also have a velocity in the opposite direction.
Now, let's find the final velocity of the car using the formula:
Total mass of cannon & car system × final velocity = 0
(500 kg + 11,000 kg) × final velocity = 0
11,500 kg × final velocity = 0
Therefore, the final velocity of the car is 0 m/s. This means that the car does not move after all the cannonballs come to rest on the right side.
Hence, the greatest distance the car can have moved from its original position is 0 meters.