How is it possible that gravitational potential energy is negative?

Answer

The potential energy is – as the word says – energy that is in the system and can be extracted from it, it is ‘in potential’. For an attractive force like gravitation, that’s the energy you generate by falling down. The higher you are (the greater the distance r), the deeper you can fall, and so the more energy is released when you fall. The potential energy thus increases as r increases, and since it varies with (1/r), there must be a minus sign in front of the term with (1/r).

In general, the sum of kinetic and potential energy is constant, the total energy. If a planet orbits the sun on an eccentric ellipse, it moves faster the closer it is to the sun, so its kinetic energy is greater, and its potential energy must be less. So again greater potential energy for greater distance, and thus with a negative sign if it goes according to 1/r.

If you absolutely have something against ‘negative energy’, then you are free to add a large constant to the potential energy such that it always remains positive and energy conservation still applies. But that makes it unnecessarily complicated.