Cannon in Box A cannon and a supply of cannonballs are inside a sealed railroad

Question

Cannon in Box

A cannon and a supply of cannonballs are inside a sealed railroad car of length L = 12.6 m, as in the figure. The cannon fires to the right; the car recoils to the left. Fired cannonballs remain in the car after hitting the far wall and landing on the floor there.

After the cannonballs have been fired, what is the greatest distance the car can have moved from its original position?

What is the speed of the car just after all the cannonballs have been fired and have landed at the end of the car?

Explanation / Answer

When the cannon is fired, the momentum of the ball is matched by the momentum of the car plus cannon and all but one of the balls. Let the muzzle velocity be Vm and the recoil velocity of the car be Vr (Mt is the total mass of the car+cannon+all the balls):

Mb*Vm = (Mt - Mb)*Vr; the recoil velocity of the car is then

Vr = Vm*Mb/(Mt - Mb)

When the ball hits the other wall, all the momentum is returned to the car and its motion stops. The distance the car moves d = Vr*t, where t is the time it takes for the ball to cross the car. The velocity of the ball relative to the car is Vm+Vr, so if L is the length of the car, t = L/(Vm+Vr). Then d = Vr*L/(Vm+Vr). Substituting for Vr from above:

d = Vm*Mb/(Mt - Mb)*L / [Vm + Vm*Mb/(Mt - Mb)]

Vm cancels out leaving

d = Mb/(Mt - Mb)*L / [1+ Mb/(Mt - Mb)]
d = Mb*L/(Mt - Mb + Mb)
d = L*Mb/Mt

This is for each cannonball fired; for n balls,

d = n*L*Mb/Mt

n = 79, Mt = 500 + 16000 + 79*43.5 = 19936.5 k. Mb = 43.5 kg, L = 79 m

d = 13.6 m

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