Grade 12 Physics · Leaving for good
The negative well
score 0

The negative well

Gravitational potential energy near a surface is U=mghU = mghU equals m g h, with mm the mass in kilograms, gg the local field strength, and hh the height above a zero you chose yourself (the bench, the floor, the ground). That chosen zero is fine while g barely changes. Leave the surface and it fails, so astronomy uses the honest form: U=GMmrU = -\dfrac{GMm}{r}, with MM the central mass, mm the orbiting mass, rr their separation in metres, and the zero fixed at infinite separation.

Hence the minus sign, which alarms everyone once and then never again: zero is infinitely far away, so everything gravity still holds sits BELOW zero. Negative means bound. Climb outward and UU rises toward zero; reach zero with nothing to spare and you have just barely escaped. Note also the single rr downstairs — the force has r2r^2, the energy has rr, and mixing them is the classic slip.

Put that at exactly zero total energy and you get v=2GMrv = \sqrt{\dfrac{2GM}{r}}, the escape speed: the launch speed from distance rr that arrives at infinity with nothing left. Set it beside the orbital speed GM/r\sqrt{GM/r} and only a factor of 21.41\sqrt{2} \approx 1.41 separates them. Orbiting is falling forever; escaping costs 41% more speed and buys the entire universe. And notice the escaping object's own mass is nowhere in it — a pebble and a spacecraft need exactly the same number.