An 80.9 kg astronaut is taking a space walk to work on the engines of his ship, which is drifting through space with a constant velocity. The astronaut, wishing to get a better view of the Universe, pushes against the ship and much later finds himself 30.3 m behind the ship. Without a thruster or tether, the only way to return to the ship is to throw his 0.504 kg wrench directly away from the ship. If he throws the wrench with a speed of 19.8 m/s relative to the ship, how long does it take the astronaut to reach the ship?
To find out how long it takes for the astronaut to reach the ship, we can use the concept of conservation of momentum.
First, let's calculate the initial momentum of the astronaut and the wrench system. The momentum (p) is given by the equation p = mass × velocity.
For the astronaut:
Initial momentum astronaut = mass astronaut × velocity astronaut
= 80.9 kg × 0 m/s (since the astronaut is initially at rest)
For the wrench:
Initial momentum wrench = mass wrench × velocity wrench
= 0.504 kg × 19.8 m/s
According to the law of conservation of momentum, the total initial momentum should be equal to the total final momentum. Therefore:
Initial momentum astronaut + Initial momentum wrench = Final momentum astronaut + Final momentum wrench
Since the astronaut and the wrench are initially at rest relative to the spaceship, the total initial momentum is 0.
0 + 0.504 kg × 19.8 m/s = (80.9 kg + 0.504 kg) × V
Where V is the final velocity of the astronaut.
Simplifying the equation:
9.9876 kg·m/s = 81.404 kg × V
V = 0.1226 m/s
Now, to calculate the time taken, we can use the equation:
Time = Distance ÷ Velocity
Since the astronaut is 30.3 m behind the ship and the velocity is 0.1226 m/s, we have:
Time = 30.3 m ÷ 0.1226 m/s
Simplifying:
Time = 247.15 seconds
Therefore, it takes approximately 247.15 seconds (or about 4 minutes and 7 seconds) for the astronaut to reach the ship.