A space vehicle is traveling at 4100 km/h relative to Earth when the exhausted rocket motor is disengaged and sent backward with a speed of 76 km/h relative to the command module. The mass of the motor is four times the mass of the module. What is the speed (in km/h) of the command module relative to Earth just after the separation?

Try applying the law of conservation of momentum in the coordinate sytem of the Earth. The momentum lost by the ejection of the motor must equal the momentum gained by the remaining command module. This assumes that the ejection occured due to a "push" of some kind from the command module -- not a rocket motor retrofiring after separation.

To find the speed of the command module relative to Earth just after the separation, we can use the principle of conservation of momentum.

The momentum of an object is given by the mass of the object multiplied by its velocity. The momentum of the system (consisting of the command module and the rocket motor) before the separation is equal to the momentum of the system after the separation.

Let's denote the mass of the command module as M and the mass of the rocket motor as 4M (since it is four times the mass of the module).

The initial momentum of the system before separation is:
M * 4100 km/h (for the command module) + 4M * (-76 km/h) (for the rocket motor moving backward)

The final momentum of the system after separation is:
M * V cm (for the command module) + 4M * (-V rm) (for the rocket motor moving backward)

Since the momentum is conserved, we can set the initial momentum equal to the final momentum:

M * 4100 km/h + 4M * (-76 km/h) = M * V cm + 4M * (-V rm)

Simplifying the equation, we get:

4100 M - 304M = V cm - 4V rm

3796M = V cm - 4V rm

Now, we can use the fact that the speed of the rocket motor relative to the command module is 76 km/h:

V rm = 76 km/h

Plugging in this value, we get:

3796M = V cm - 4 * 76 km/h

Simplifying further:

3796M = V cm - 304 km/h

Rearranging the equation:

V cm = 3796M + 304 km/h

The mass of the rocket motor (4M) is given, but the mass of the command module (M) is not provided. Therefore, we cannot determine the exact speed of the command module relative to Earth just after the separation without knowing the value of M.