The speed of the command module relative to the Earth just after separation is 4955 km/h.
To solve this problem, we can use the law of conservation of momentum. According to this law, the momentum before the separation is equal to the momentum after the separation. The momentum of an object is given by the product of its mass and velocity.
Let's denote the mass of the command module as M. The mass of the exhausted rocket motor is 4M. The initial velocity of the space vehicle relative to the Earth is 5275 km/h, and the velocity of the rocket motor relative to the command module is 80 km/h.
Using the law of conservation of momentum, we can set up the equation:
(M x 5275 km/h) = (4M x 80 km/h) + (M x V)
where V is the final velocity of the command module relative to the Earth.
Simplifying the equation, we get:
5275M = 320M + MV
Dividing both sides by M, we have:
5275 = 320 + V
Subtracting 320 from both sides, we get:
V = 5275 - 320 = 4955 km/h
Therefore, the speed of the command module relative to the Earth just after the separation is 4955 km/h.