Answer to Question #250973 in Physics for Pranali

Question

A 500 kg cannon and a supply of 33 cannon balls, each with a mass of 16.2 kg, are inside a sealed railroad car with a mass of 33000 kg and a length of 48 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?

Answer 1

Initially, the speed of the center of mass is 0. The position of the center of mass is

$$x_c=\frac{mx_1+Mx_2}{m+M}.$$

In the equation above, m is the mass of the cannon ball and M is the mass of the car and cannon.

The center of mass does not change because there is no external force, so:

$$m\Delta x_1+M\Delta x_2=0.$$

Since the balls hit the wall, their maximum displacement will be length of the car L minus the recoil distance D:

$$\Delta x_2=L-D$$

Substitute this in the second equation:

$$m(L-D)-MD=0,\\\space\\ D=\frac{mL}{M+m}=\frac L{1+M/m}.$$

As we see, the car cannot move further than the length of the car.

Substitute our values:

$$D=\frac {48}{1+(33000+500)/(33·16.2)}=0.75\text{ m}.$$

The speed of the car as all balls are fired is zero.