Gravitational Potential and EnergyGravitational Potential and Energy

Gravitational potential energy is the work done in bringing a mass from infinity to the point. It is found by integration – it’s equal to the area under a force-distance graph. For a point or spherical mass, it’s given by

$E=mV_{g}=-frac{GM}{r}$

Gravitational potential is the GPE per unit mass – the work done in bringing a unit mass from infinity to the point. Gravitational potential at infinity is zero, so gravitational potentials elsewhere are always negative. It is given by:

$v_{g}=-frac{GM}{r}$

Changes in gravitational potential are derived by subtracting one gravitational potential from another. Moving towards the parent object decreases gravitational potential, whist moving away increases it.

Escape velocity is the minimum velocity for a projectile to leave a gravitational field. It is found by equating KE to GPE.

$frac{mv^{2}}{2}=frac{GMm}{r}$

Divide by m on both sides, this gets rid of all mentions of m. That means the final answer will not depend on the mass leaving the planet at al

$frac{v^{2}}{2}=frac{GM}{r}$

Rearrange:

$v=sqrt{frac{2GM}{r}}$