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Chapter 7: Problem 18

A railway flat car is rushing along a level frictionless track at a speed of \(45 \mathrm{~m} / \mathrm{s}\). Mounted on the car and aimed forward is a cannon that fires \(65-\mathrm{kg}\) cannon balls with a muzzle speed of \(625 \mathrm{~m} / \mathrm{s}\). The total mass of the car, the cannon, and the large supply of cannon balls on the car is \(3500 \mathrm{~kg}\). How many cannon balls must be fired to bring the car as close to rest as possible?

Short Answer

Expert verified
Four cannonballs must be fired to bring the car as close to rest as possible.

Step by step solution

01

Understand the initial momentum

The initial momentum of this system can be calculated by multiplying the total mass of the system by their initial velocity. Hence, initial momentum = \( mass_{total} × velocity_{initial} = 3500 kg × 45 m/s = 157500 kg·m/s \).
02

Understand the final momentum

If we define forward as the positive direction, we can see that the final momentum of the system, following firing a cannonball, is composed of the momentum of the car-cannon system and the cannonball. Hence, it can be defined as: Final momentum = \( mass_{car-cannon} × velocity_{car-cannon} ) + ( mass_{cannonball} × velocity_{cannonball} ) . \) Assuming that the car comes as close to rest as possible, we can set its final forward velocity to 0, hence, Final momentum = \( 0 + ( 65 kg × -625 m/s ) = -40625 kg·m/s \), where we've taken the velocity of the ball as negative since it is going in the negative or backwards direction with respect to the car.
03

Evoke the Conservation of Momentum Law

According to the conservation of momentum, the total initial momentum should equal to the total final momentum. Setting these equal we get: \( 157500 kg·m/s = n × -40625 kg·m/s\). Solve for n to get number of cannonballs.
04

Compute the number of cannonballs

Solving for n we get: \( n = 157500 kg·m/s / -40625 kg·m/s = -3.88 \), n should be a whole number, since we cannot fire a fraction of cannonball. From the context, the negative sign means cannonballs are fired in the opposite direction of the initial motion of the car, thus we take the absolute value and then round up. Hence, the number of cannonballs to be fired is 4.

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