An 94.4 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 29.6 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 0.540 kg, and the astronaut throws the wrench with a speed of 21.0 m/s, how long does it take the astronaut to reach the ship?

To solve this problem, we need to analyze the motion of the astronaut and the wrench separately and then find the time it takes for the astronaut to reach the ship.

Let's start by analyzing the motion of the wrench. Since no external forces are acting on the wrench, we can apply the principle of conservation of momentum. The initial momentum of the wrench is zero since it was at rest relative to the spaceship. When the astronaut throws the wrench, she imparts a momentum to it in the opposite direction of her motion. We can use the equation:

m1 * v1 = m2 * v2

Where m1 and v1 are the mass and velocity of the astronaut, m2 and v2 are the mass and velocity of the wrench.

Plugging in the values:
(94.4 kg)(0 m/s) = (0.540 kg)(21.0 m/s)

Simplifying the equation:
0 = 11.34 kg*m/s

This tells us that the momentum gained by the wrench is equal in magnitude but opposite in direction to the initial momentum of the astronaut.

Now, let's analyze the motion of the astronaut. We can use the equation for velocity to calculate the time it takes for the astronaut to reach the ship:

velocity = distance / time

The distance the astronaut needs to cover is the same as the distance the wrench traveled behind the ship, which is given as 29.6 m. We know the astronaut's initial velocity is zero, and we can assume her final velocity when she reaches the ship is the same as the velocity of the spaceship.

Rearranging the equation, we get:

time = distance / velocity

time = 29.6 m / 0 m/s (since the spaceship is moving with a constant velocity)

Since we are dividing by zero, we cannot directly calculate the time using this equation. However, we can use the conservation of momentum to find the final velocity of the spaceship (and hence the astronaut). Since the wrench gained momentum, the spaceship (and the astronaut) must have lost an equal amount of momentum. We can use the equation:

m1 * v1 = m2 * v2

Where m1 and v1 are the mass and initial velocity of the spaceship (and the astronaut), and m2 and v2 are the mass and final velocity of the spaceship (and the astronaut).

Plugging in the values:
(94.4 kg)(0 m/s) = (94.4 kg + 0.540 kg) * v2

Simplifying the equation:
0 = (94.4 kg)(v2)

From this equation, we can see that the final velocity of the spaceship is zero. Since the spaceship is stationary, it does not move away from the astronaut during the time of flight of the wrench.

Therefore, the time it takes for the astronaut to reach the ship is infinite since the spaceship does not move away from the astronaut.