A monkey is climbing on a rope that passes over a smooth light...

Let the mass of the monkey and the block be $m$.

Let $T$ be the tension in the rope. Since the pulley is smooth and light, the tension remains the same on both sides of the rope.

The forces acting on the block are the tension $T$ upwards and its weight $mg$ downwards. The equation of motion for the block is:

$T - mg = m a_b$

where $a_b$ is the upward acceleration of the block.

The forces acting on the monkey are the tension $T$ upwards (since the force the monkey exerts on the rope equals the tension) and its weight $mg$ downwards. The equation of motion for the monkey is:

$T - mg = m a_m$

where $a_m$ is the upward acceleration of the monkey.

From the two equations, we get:

$m a_b = m a_m \Rightarrow a_b = a_m$

Since both the monkey and the block start from rest and have the same acceleration, their velocities and displacements will be equal at all times. Therefore, they will move in the same direction with equal acceleration, and the distance between them will remain constant as time passes.