The overall theme for this question is that in the absence of any external influences, total momentum (as well as total internal energy) is always conserved. The basic principle behind getting both astronauts back to the ISS is for the first astronaut (5 yards away from the ISS) to rescue...
The overall theme for this question is that in the absence of any external influences, total momentum (as well as total internal energy) is always conserved. The basic principle behind getting both astronauts back to the ISS is for the first astronaut (5 yards away from the ISS) to rescue the second astronaut (10 yards from the ISS) before they both can return to the ISS together.
A) If we consider the system of reference to be the ISS, Astronaut 1, and Astronaut 2, then both astronauts posses 0 momentum as they are both floating alongside the ISS, neither moving towards or away from it in any direction. Astronaut 1 should take one of the tools from their belt and throw it in the opposite direction as Astronaut 2. This will impart momentum (inertia in motion) to the tool as it will begin moving towards the ISS. However, according to Newton's 3rd Law, Astronaut 1 will now possess that same amount of momentum, but in the opposite direction (towards Astronaut 2). In short, this is like an elastic collision, though the total momentum before was 0. According to the momentum equation, the larger mass of the astronaut will result in a much smaller velocity than the tool of choice would have received. Astronaut 1, depending on the availability of tools, can continue this process until they reach Astronaut 2. Once they collide, Astronaut 2 will need to catch Astronaut 1 and hold on tight, resulting in an inelastic collision. The result will be Astronaut 1 slowing down as Astronaut 2 will have, in essence, increased his/her mass. The original process of discarding tools should be repeated, though this time opposite the ISS. As each tool is thrown away from the station, the astronaut pair will slow down, stop, and eventually achieve a velocity in the direction of the ISS.
B) The conservation of energy in this action is larger than simply throwing the hammer. The Law of Conservation of Energy states that the total internal energy of a system remains constant, but it can be transferred between objects and transformed to different types of energy. The astronaut has an amount of potential energy stored inside of their bodies. The action of throwing the hammer will utilize that chemical potential energy and convert it into kinetic energy, enabling the astronaut to move. That motion will then cause the hammer to move, imparting kinetic energy to the hammer. Overall, the total energy of the astronaut and the hammer remains constant, but the total energy of the astronaut decreases while the energy of the hammer increases. Potential energy of the astronaut is converted to kinetic energy and then transferred as kinetic energy to the hammer.
C) When bodies rotate, inertia in motion is referred to as angular momentum. Just as in a linear system, angular momentum is also conserved during encounters. When the astronaut throws the hammer with a clockwise rotation, he imparts angular momentum on the hammer. Before he threw it, the total rotational momentum of the hammer and he combined was 0. After he throws it, the total rotational momentum will still be 0. As a result, he will be rotating in the opposite direction as the hammer, but with a much slower rotation due to a larger mass.