Final answer:
When an astronaut throws a wrench in space, they experience a recoil force in the opposite direction. The astronaut's recoil speed can be calculated using the principle of conservation of momentum. By substituting the given values into the momentum equation, we find that the astronaut's recoil speed is -0.22 m/s.
Step-by-step explanation:
According to Newton's third law of motion, for every action, there is an equal and opposite reaction. In this case, when the astronaut throws the wrench, they exert a force on the wrench in one direction, causing the wrench to move with a certain velocity. As a result, the astronaut experiences a recoil force in the opposite direction, causing them to move with a certain speed.
To find the astronaut's recoil speed, we can use the principle of conservation of momentum. The momentum of the astronaut before throwing the wrench is zero since they are at rest. The momentum of the astronaut and the wrench system after the throw should also be zero since there is no external force acting on them. Therefore, the momentum of the astronaut and the wrench after the throw can be given as:
0 = (mass of the astronaut) * (recoil speed) + (mass of the wrench) * (wrench velocity)
Substituting the given values (mass of the astronaut = 80 kg, mass of the wrench = 0.75 kg, wrench velocity = 24 m/s), we can solve for the recoil speed of the astronaut:
(80 kg) * (recoil speed) + (0.75 kg) * (24 m/s) = 0
Solving this equation, we find that the astronaut's recoil speed is -0.22 m/s. The negative sign indicates that the astronaut moves in the opposite direction of the thrown wrench.