An astronaut of mass 78.0 kg is taking a space walk to work on the International Space Station. Because of a malfunction with the booster rockets on his spacesuit, he finds himself drifting away from the station with a constant speed of 0.500 m/s. With the booster rockets no longer working, the only way for him to return to the station is to throw the 8.55 kg wrench he is holding.
A) He throws the wrench with speed 12.65 m/s WITH RESPECT TO HIMSELF. After he throws the wrench, how fast is the astronaut drifting toward the space station?
B) What is the speed of the wrench with respect to the space station?
To answer these questions, we need to apply the principle of conservation of momentum. According to this principle, the total momentum before an event is equal to the total momentum after the event, as long as no external forces act on the system.
First, let's calculate the initial momentum of the astronaut and the wrench before he throws it. The momentum of an object is given by the product of its mass and velocity.
Given:
Mass of the astronaut (m1) = 78.0 kg
Speed of the astronaut (v1) = 0.500 m/s
Mass of the wrench (m2) = 8.55 kg
Speed of the wrench (v2, relative to the astronaut) = 12.65 m/s
A) To find how fast the astronaut is drifting toward the space station after he throws the wrench, we need to calculate the final velocity (vf) of the astronaut.
Using the conservation of momentum, we can set up the equation:
Initial momentum of the system = Final momentum of the system
(mass of astronaut * initial velocity of astronaut) + (mass of wrench * initial velocity of wrench) = (mass of astronaut * final velocity of astronaut) + (mass of wrench * final velocity of wrench)
(78.0 kg * 0.500 m/s) + (8.55 kg * 12.65 m/s) = (78.0 kg * vf) + (8.55 kg * 0 m/s)
Simplifying the equation, we get:
39 kg * m/s + 108.0075 kg * m/s = 78 kg * vf
147.0075 kg * m/s = 78 kg * vf
Dividing both sides by 78 kg, we find:
1.884 kg * m/s = vf
Therefore, the astronaut is drifting toward the space station at a speed of 1.884 m/s.
B) Next, let's calculate the speed of the wrench with respect to the space station (v_wrench_station).
To find the speed of the wrench with respect to the space station, we need to consider the relative velocity between the astronaut and the space station.
Relative velocity (v_relative) = (velocity of the astronaut) + (velocity of the wrench relative to the astronaut)
v_relative = 0.500 m/s + 12.65 m/s = 13.15 m/s
Therefore, the speed of the wrench with respect to the space station is 13.15 m/s