A 87.2 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 44.2 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. The wrench has a mass of 0.411 kg, and the astronaut throws the wrench with a speed of 24.1 m/s.

How long does it take the astronaut to reach the ship?
please show the steps

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To solve this problem, we can utilize the principle of conservation of momentum. The momentum before the wrench is thrown is equal to the momentum after the wrench is thrown in the opposite direction.

Step 1: Calculate the momentum before the wrench is thrown:
The momentum of the astronaut before the wrench is thrown can be calculated using the formula: p = m * v
where p represents momentum, m represents mass, and v represents velocity.
The mass of the astronaut is 87.2 kg, and since the astronaut is at rest relative to the spaceship, the velocity is 0 m/s. Thus, the momentum before the wrench is thrown is 0 kg*m/s.

Step 2: Calculate the momentum after the wrench is thrown:
The momentum of the wrench after it is thrown can be calculated in the same way as before.
The mass of the wrench is 0.411 kg, and the velocity at which it is thrown is 24.1 m/s in the opposite direction. Thus, the momentum after the wrench is thrown is -9.93651 kg*m/s.
Note: The negative sign indicates that the momentum is in the opposite direction of the spaceship's velocity.

Step 3: Apply the principle of conservation of momentum:
According to the principle of conservation of momentum, the total momentum before and after the wrench is thrown should be equal. So we can set up the equation:
Momentum before = Momentum after
0 kg*m/s = -9.93651 kg*m/s

Step 4: Solve for the time taken by the astronaut to reach the ship:
To find the time taken by the astronaut to reach the ship, we need to consider the change in position of the astronaut while the spaceship is drifting at a constant velocity.
The distance covered by the astronaut (44.2 m) is given, and the astronaut's mass (87.2 kg) is known. We can use the equation:
Distance = Velocity * Time
Rearranging the equation, we get:
Time = Distance / Velocity

Plugging in the values:
Time = (44.2 m) / (24.1 m/s)
Time ≈ 1.832365 m/s

Therefore, it takes approximately 1.832365 seconds for the astronaut to reach the ship.