A Newtonian problem I tried to imagine what would happen to a theoretical astronaut in the following thought experiment. The astronaut floats in space with the velocity of zero. He/she has a 10kg object and a wall 2 meters away (it floats in one place too). The wall and the object both have a(n) elasticity (coefficient of restitution) of 0.99. What exactly happens when the astronaut throws the object towards the wall? Because in my interpretation, when the object is tossed away with the force of x, the astronaut accelerates in the opposite direction with the same amount of impulse. And when it collides with the wall it bounces back, the wall pushes a bit and the object bounces back towards the astronaut. When he/she catches it again, the momentum of it accelerates a bit again on the person. And the object gains upon because its mass is like 10% the astronaut and it has more speed. Please help me solve this problem because I cannot get a good night's sleep until I figure this one out. Thanks in advance for pointing out any errors in my model.
 A: As @gandalf61 noted that there is no external force the center of the mass stays still till the object and the astronaut hits to the wall. Then because the collisions are not fully elastic there will be some loss in energy and also momentum (between astronout and the object). Considering the amount of defect center of the mass can change but for the given elasticity coefficient collision will not alter the result as you interpreted in the answer.
A: There are no external forces acting on the total system of astronaut, ball and wall. So if the centre of mass of the system is initially at rest (e.g in a reference frame in which the astronaut and the wall are both initially at rest) then it will remain at rest.
So after the astronaut has thrown the ball, it has bounced off the wall, and the astronaut has caught the ball again then the momentum of the wall will be equal and opposite to the momentum of the astronaut and ball. In other words the wall and the astronaut/ball will be moving in opposite directions with speeds that are inversely proportional to their masses.
