Question

the astronaut realizes that the only way to get back to the space station is to throw the tool - kit.
(c) identify the direction the astronaut must throw the tool - kit: toward the space station or away from it. use the law of conservation of momentum to explain how throwing the tool - kit may return the astronaut to the space station.

Explanation:

Step1: Recall conservation of momentum

The total initial momentum of the astronaut - tool - kit system is zero (assuming they are at rest relative to each other initially in space). According to the law of conservation of momentum, the total final momentum must also be zero, i.e., $p_{total - initial}=p_{total - final}=0$.

Step2: Determine direction

The astronaut must throw the tool - kit away from the space - station. Let the momentum of the tool - kit be $p_{tool}$ and the momentum of the astronaut be $p_{astro}$. Since $p_{total - final}=p_{tool}+p_{astro} = 0$, then $p_{astro}=-p_{tool}$. When the astronaut throws the tool - kit away from the space - station, the astronaut will gain a momentum in the opposite direction (towards the space - station) due to the conservation of momentum.

Answer:

The astronaut must throw the tool - kit away from the space - station. Throwing the tool - kit away from the space - station gives the tool - kit a momentum in that direction. By the law of conservation of momentum, the astronaut gains an equal and opposite momentum, which propels the astronaut towards the space - station.