Let’s tie this all together. it makes sense that, if the rope force remains greater than the gravitational force, the child keeps speeding up; and if the rope force becomes less than the gravitational force, the child slows down. by this line of intuitive reasoning, what happens to the child’s motion if the rope force equals the child’s weight, i. e., if the rope force "compromises" between being greater than and being less than the child’s weight? explain.

If child weight is equal to rope force then child will move with uniform speed

or we can say that the child will remain at rest in his position

Explanation:

As we know that child is hanging by rope

so here there will be two forces on the child

1) Weight or gravitational force which act vertically downwards

2) Tension in the rope which act vertically upwards

Now if child will accelerate upwards then tension force must be more than the weight of the child

If tension force is less than the weight then child will decelerate and his speed will decrease

if tension force is equal to child weight then in that case the child will remain at rest or it will move with same speed

not too sure thinking it will always be directed towards the centre of the earth. could be wrong not sure