Gravitational potential energy — the honest formula
Near the ground we happily write gravitational potential energy as U = mgh. But that only works when g is roughly constant, i.e. over heights small compared with the whole planet. Once you move far out — to satellites, the Moon, other planets — g itself weakens with distance, and we need the true gravitational potential energy.
The gravitational potential energy of two masses a distance r apart, taken as zero when they are infinitely far apart.
Why the minus sign? We choose U = 0 at infinite separation, where the masses feel nothing. Bringing them closer, gravity does positive work, so the energy drops below zero. A negative total energy therefore means the two masses are bound together — you would have to add energy to pull them apart. Hold that thought; it is the key to escape.
The total energy of an orbit
An orbiting body carries kinetic energy K = \tfrac{1}{2}mv^{2} and potential energy U = -GMm/r. Add them. Using the circular-orbit speed v = \sqrt{GM/r} from Guide 3, the kinetic energy is K = GMm/2r, and the total comes out neat and negative:
Total mechanical energy of a circular orbit. It is negative — the orbit is bound. (For a general ellipse, replace r with the semi-major axis a.)
Worked example: escape velocity
To just barely escape a planet, launch with exactly enough kinetic energy to climb the potential-energy hill all the way to infinity — that is, set the total energy to zero. Starting from radius r:
Set kinetic energy equal to the depth of the potential well and solve. Note v_esc = √2 × the circular-orbit speed at the same radius.
For the Earth's surface, GM \approx 3.99\times 10^{14}\ \mathrm{m^{3}/s^{2}} and r = R_\oplus \approx 6.37\times 10^{6}\ \mathrm{m}, which give v_{\text{esc}} \approx 1.12\times 10^{4}\ \mathrm{m/s} — about 11.2 km/s, or roughly 40,000 km/h. This is the escape velocity from Earth.
Tides: gravity that stretches
The tidal (stretching) force scales as 1/r³ — much steeper than gravity's own 1/r². Here Δr is the size of the stretched body and r its distance from the mass M.
That 1/r^{3} dependence explains a puzzle: the Sun is 27 million times more massive than the Moon, yet its tides are only about half as strong, because it is so much farther away that the cube of the distance overwhelms the extra mass. When Sun and Moon line up (new and full Moon) their bulges add into large spring tides; at right angles (half Moon) they partly cancel into gentle neap tides. Tidal forces also crack moons apart, heat the volcanic surface of Jupiter's moon Io, and lock the Moon so it always shows us the same face.
Where this leads: beyond Newton
Newton's gravity is astoundingly accurate — it steers spacecraft to distant planets to the second. But it is not the final word. The orbit of Mercury precesses by a tiny amount that Newton's law cannot fully account for. In 1915 Einstein's general relativity reinterpreted gravity not as a force reaching across space, but as the curvature of spacetime itself: mass tells spacetime how to curve, and curved spacetime tells matter how to move.
General relativity nails Mercury's precession, bends starlight, and slows clocks in strong gravity. Its boldest prediction — ripples in spacetime called gravitational waves, shed by colliding black holes — was finally detected in 2015, a century after Einstein wrote it down. That is where this track hands off: to relativity, to astrophysics, and to the study of black holes, where the gentle inverse-square law you have just mastered becomes the doorway to the most extreme physics in the universe.