An athlete grips a light rope that passes over a low-friction pulley attached to the ceiling of a gym. a sack of sand precisely equal in weight to the athlete is tied to the other end of the rope. both the sand and the athlete are initially at rest. the athlete climbs the rope, sometimes speeding up and slowing down as he does so. what happens to the sack of sand? explain your answer.

The sack of sand will move the same distance than the athlete and in the same direction towards the pulley. So the distance between the athlete and the sack of sand will not change.

Explanation:

there are 2 forces acting over the athlete and the sack of sand, the tension of the rope and their own weight, since the weight of the athlete and the sack of sand are the same the system is in equilibrium and none of them will move until they break the equilibirium, to do that,  the athlete pulls the rope downwards, so he is exerting a force over the rope and by third Newton law the rope will exert the same force in the oposite direction over the athlete, this means that the tension of the rope will increase which means that T>m*g and the athelete will have an acceleration upwards, and the same happens to the sack of sand, so they have the same acceleration upwards and they move the same distance in the same direction since they have the same mass.