A space vehicle is traveling at 5260 km/h relative to Earth when the exhausted rocket motor is disengaged and sent backward with a speed of 89 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?
To solve this problem, we can apply the principle of conservation of momentum.
Conservation of momentum states that the total momentum of an isolated system remains constant before and after an event, as long as no external forces act on the system. In this case, the space vehicle and the rocket motor form an isolated system.
Let's break down the problem and gather the necessary information:
1. The speed of the space vehicle relative to Earth = 5260 km/h.
2. The speed of the rocket motor relative to the command module = 89 km/h.
Let's assume the mass of the command module is M, and the mass of the rocket motor is 4M (four times the mass of the module).
According to the law of conservation of momentum, the momentum before the separation should be equal to the momentum after the separation.
Momentum (before) = Momentum (after)
The momentum of an object is given by the product of its mass and velocity.
M x Velocity (vehicle relative to Earth) = (4M) x (Velocity (command module relative to Earth) - Velocity (rocket motor relative to the command module))
The velocity of the vehicle relative to Earth is already given as 5260 km/h, and the velocity of the rocket motor relative to the command module is given as 89 km/h.
Let's substitute these values into the equation:
M x 5260 = 4M x (Velocity (command module relative to Earth) - 89)
Now, solve for Velocity (command module relative to Earth):
5260 = 4 x (Velocity (command module relative to Earth) - 89)
Divide both sides by 4:
1315 = Velocity (command module relative to Earth) - 89
Add 89 to both sides:
Velocity (command module relative to Earth) = 1404 km/h
Therefore, the speed of the command module relative to Earth just after the separation is 1404 km/h.