Question

If you jump on the moon you will rise much higher than if you jump on the earth

Answer 1

the height of your jump only depends upon the value of g as because the gravitational pull is the only force which leads you back to the ground after the jump.
in case of moon and earth the gravitational constant g is in the ratio 1:6 so the gravitational force experienced on earth is 6 times greater than on the moon.
so if u jump on moon surface the rise is higher.

Answer 2

in the Earth man is pushing against the interia of his mass and the downward force of his Earth weight
in the moon is pushing against the interia of his mass and downward force of his moon weight
clearly the moon man will have a higher velocity when his feet leave the ground than the Earth man and so will travel over six times higher how much higher will depend on how long the acceleration zone is