Potential energy $= -GMm/r$ then how could potential energy ever be positive?

Question

Using the $U = -GMm/r$ where $r$ is a distance which would be positive how could potential energy ever be positive?

How could it be positive according to the equation $mgh$?

Answer 1

It does not need to be positive. In physics we only care about potential energy differences. The first formula the OP quotes already has a chosen reference at infinity. With such a reference point gravitational potential energy will always be negative.

The $mgh$ formula is only a reduced version where differences are approximated linearly for gravitational fields that do not vary too much within the heights of interest. There one is free to choose any useful reference such as the ground in problems of classical mechanics.

Answer 2

It could become zero as r tends to infinity , positive potential energy indicated system is not bounded which is not possible unless r tends to infinity

Answer 3

It couldn't.

Remember that potential energy is a value based on an arbitrary reference. The formula

$$U_g=-G \frac{mM}{r}$$

is based on a zero-level at inifinity. Thus the value will never be positive since we can't reach beyond that. The formula

$$U_g=mgh$$

is based on a zero level at ground level (or you could choose it differently). Thus the value can be positive if we reach a value higher than that.

Potential energies can be calculated for arbitrarily chosen references because only their differences matter. We shouldn't care about the value itself but only about the difference between the value in one scenario compared to the value in another scenario.