An 85.4 kg astronaut is working on the engines of a spaceship that is drifting through space with a constant velocity. The astronaut turns away to look at Earth and several seconds later is 22.4 m behind the ship, at rest relative to the spaceship. The only way to return to the ship without a thruster is to throw a wrench directly away from the ship. If the wrench has a mass of 0.370 kg, and the astronaut throws the wrench with a speed of 16.0 m/s, how long does it take the astronaut to reach the ship?
To solve this problem, we can use the concept of conservation of momentum. The total momentum of a system remains constant if no external forces act on it.
Let's break down the problem and analyze the situation:
1. The astronaut has a mass of 85.4 kg.
2. The wrench has a mass of 0.370 kg.
3. The astronaut throws the wrench with a speed of 16.0 m/s.
4. The astronaut wants to return to the ship by throwing the wrench away.
5. The astronaut ends up at rest relative to the spaceship.
First, let's calculate the momentum of the astronaut before throwing the wrench:
Momentum of the astronaut = mass of astronaut * velocity of astronaut
Momentum of the astronaut = 85.4 kg * 0 m/s (since the astronaut is initially at rest)
Next, let's calculate the momentum of the wrench before being thrown:
Momentum of the wrench = mass of wrench * velocity of wrench
Momentum of the wrench = 0.370 kg * 16.0 m/s
Now, since momentum is conserved in this isolated system, the total momentum before throwing the wrench must be equal to the total momentum after throwing the wrench.
Total momentum before = Total momentum after
Momentum of the astronaut + Momentum of the wrench before = Momentum of the astronaut after + Momentum of the wrench after
0 + (0.370 kg * 16.0 m/s) = (85.4 kg + 0.370 kg) * V (where V is the velocity of the astronaut and the wrench after the throw)
Simplifying the equation:
0.370 kg * 16.0 m/s = 85.77 kg * V
5.92 kg·m/s = 85.77 kg * V
V = 5.92 kg·m/s / 85.77 kg
Now, we know that the time it takes for the astronaut to reach the ship is given by the distance traveled divided by the velocity of the astronaut. The astronaut travels a distance of 22.4 m.
Time = Distance / Velocity
Time = 22.4 m / (5.92 kg·m/s / 85.77 kg)
Calculating the time:
Time = 22.4 m / 0.069 m/s
Finally, we can calculate:
Time = 324.64 seconds
Therefore, it takes approximately 324.64 seconds for the astronaut to reach the ship after throwing the wrench.