An astronaut of mass 84.0 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 0.570 m/s. With the booster rockets no longer working, the only way for him to return to the station is to throw the 7.85 kg wrench he is holding.
Which way should he throw the wrench?
away from the station?
toward the station?
He throws the wrench with speed 16.67 m/s WITH RESPECT TO HIMSELF.
After he throws the wrench, how fast is the astronaut drifting toward the space station?
What is the speed of the wrench with respect to the space station?
To solve this problem, we can use the principle of conservation of momentum. The total momentum before throwing the wrench should be equal to the total momentum after throwing the wrench.
1. First, let's find the initial momentum of the astronaut and the wrench before throwing the wrench.
The momentum (p) is given by the product of the mass (m) and the velocity (v):
momentum = mass × velocity
The initial momentum of the astronaut is:
Pastronaut = mastronaut × vastronaut,
where mastronaut = 84.0 kg (mass of the astronaut) and vastronaut = 0.570 m/s (drifting speed of the astronaut).
The initial momentum of the wrench is:
Pwrench = mwrench × vwrench,
where mwrench = 7.85 kg (mass of the wrench) and vwrench = 16.67 m/s (thrown speed of the wrench with respect to the astronaut).
2. Next, let's find the final momentum of the astronaut and the wrench after throwing the wrench.
Since there are no external forces acting on the system, the total momentum of the system (astronaut + wrench) should be conserved.
The final momentum of the astronaut and the wrench is:
Pfinal = Pafterthrowing = Pastronaut + Pwrench
3. Now, let's calculate the final velocity of the astronaut and the wrench.
The final velocity of the system is given by dividing the final momentum by the total mass of the system.
Final velocity (vfinal) = Pfinal / (mastronaut + mwrench)
Now, let's answer the specific questions:
Which way should he throw the wrench? Away from the station or toward the station?
The astronaut should throw the wrench away from the station. By throwing it in the opposite direction to the desired movement, the resulting momentum will help counteract the initial drifting speed and bring the astronaut closer to the station.
After he throws the wrench, how fast is the astronaut drifting toward the space station?
The final velocity of the astronaut (vfinal) will give us the speed at which he is drifting toward the space station.
What is the speed of the wrench with respect to the space station?
The speed of the wrench with respect to the space station can be calculated by subtracting the velocity of the astronaut from the final velocity of the wrench (relative to the astronaut).
Speed of the wrench with respect to the space station = vwrench - vastronaut.