a) The figure is below.
A person has the sensation of weight because of the normal reaction from the ground, that equals his weight. As long as the elevator falls with acceleration g downwards, the person inertia (`m*a =m*g` ) is directed upwards and cancels his weight `G.` Because the person weight is cancelled (`G-m*a =0`) there will be no normal reaction from the lift floor `N =0` on to the person and he will feel weightless.
b) The astronauts in orbit around the Earth follow a circular trajectory (within the first approximation). The weight of the astronauts that is directed toward the center of the Earth, thus it has the direction of a radius of the trajectory. This time the weight of the astronauts is canceled by the centrifugal force that appears because of the circular path. (In fact this is the condition of orbiting in a stationary path around the Earth: the centrifugal force on a mass need to equal its weight). The figure is below.
c) This is an example of a plastic collision seen backwards. Instead of two masses that collide and remain stuck together we have two masses first stuck together that separate. The total momentum with respect to the not moving laboratory frame is the same before and after separation. Before separation momentum is zero. After separation momentum is
`Mv_1+ mv_2 =0`
`v_2 = -(M/m)*v_1 =-(70/52)*1.3 =-1.75 m/s`
The figure is below.
The magnitude of the speed of the second astronaut is 1.75 m/s. The minus sign shows that the direction of the speed of the second astronaut is opposing the direction of the speed of the first astronaut.