information:classification
In space

The force of gravity exerted by a mass M on another mass m is given by F = − G M m r 2 r ^ = − G M m r 3 r, {\displaystyle \mathbf {F} =-{\frac {GMm}{r^{2}}}{\hat {\mathbf {r} }}=-{\frac {GMm}{r^{3}}}\mathbf {r},} where r is the position vector from M to m and r̂ is the unit vector in the direction of r.

Let the mass m move at the velocity v; then the work of gravity on this mass as it moves from position r (t) to r (t) is given by W = − ∫ r (t 1) r (t 2) G M m r 3 r ⋅ d r = − ∫ t 1 t 2 G M m r 3 r ⋅ v d t. {\displaystyle W=-\int _{\mathbf {r} (t_{1})}^{\mathbf {r} (t_{2})}{\frac {GMm}{r^{3}}}\mathbf {r} \cdot d\mathbf {r} =-\int _{t_{1}}^{t_{2}}{\frac {GMm}{r^{3}}}\mathbf {r} \cdot \mathbf {v} \,dt.} Notice that the position and velocity of the mass m are given by r = r e r, v = d r d t = r ˙ e r + r θ ˙ e t, {\displaystyle \mathbf {r} =r\mathbf {e} _{r},\qquad \mathbf {v} ={\frac {d\mathbf {r} }{dt}}={\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{t},} where e and e are the radial and tangential unit vectors directed relative to the vector from M to m, and we use the fact that d e r / d t = θ ˙ e t. {\displaystyle d\mathbf {e} _{r}/dt={\dot {\theta }}\mathbf {e} _{t}.} Use this to simplify the formula for work of gravity to, W = − ∫ t 1 t 2 G m M r 3 (r e r) ⋅ (r ˙ e r + r θ ˙ e t) d t = − ∫ t 1 t 2 G m M r 3 r r ˙ d t = G M m r (t 2) − G M m r (t 1). {\displaystyle W=-\int _{t_{1}}^{t_{2}}{\frac {GmM}{r^{3}}}(r\mathbf {e} _{r})\cdot \left({\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{t}\right)dt=-\int _{t_{1}}^{t_{2}}{\frac {GmM}{r^{3}}}r{\dot {r}}dt={\frac {GMm}{r(t_{2})}}-{\frac {GMm}{r(t_{1})}}.} This calculation uses the fact that d d t r − 1 = − r − 2 r ˙ = − r ˙ r 2. {\displaystyle {\frac {d}{dt}}r^{-1}=-r^{-2}{\dot {r}}=-{\frac {\dot {r}}{r^{2}}}.} The function U = − G M m r, {\displaystyle U=-{\frac {GMm}{r}},} is the gravitational potential function, also known as gravitational potential energy. The negative sign follows the convention that work is gained from a loss of potential energy.