Define the gravitational potential at a point.
For a point mass $M$, the gravitational potential $\phi$ at distance $r$ is $\phi = -\frac{GM}{r}$, where $G$ is the gravitational constant. Explain what the negative sign in this expression means.
A spherical planet may be treated as a single isolated point mass with all its mass concentrated at the centre. A small mass $m$ is moving close to, and perpendicular to, the surface of the planet. The mass moves away from the planet by a short distance $h$. State and explain why the change in gravitational potential energy $\Delta E_p$ of the mass is given by $\Delta E_p = mgh$, where $g$ is the acceleration of free fall.
In part (c), the planet has mass $M$ and a diameter of $6.8 \times 10^3\,\text{km}$. For this planet, $GM$ is $4.3 \times 10^{13}\,\text{N m}^2\,\text{kg}^{-1}$. A rock starts from rest far from the planet and speeds up towards it. Assuming that the planet has negligible atmosphere, calculate the speed of the rock when it reaches the planet’s surface.