Define the gravitational potential at a point.
A rocket is fired from the planet’s surface and travels along a radial line, as illustrated in Fig. 1.1. The planet may be modelled as an isolated sphere of radius $R$ with all its mass $M$ concentrated at its centre. Point A is $R$ from the planet’s surface. Point B is $4R$ from the surface.
Demonstrate that the gravitational potential difference $\Delta \phi$ between points A and B is $\Delta \phi = \frac{3GM}{10R}$, where $G$ denotes the gravitational constant.
The rocket motor is cut off at point A. While travelling from A to B, the rocket keeps a constant mass of $4.7 \times 10^4\,\text{kg}$ and its kinetic energy falls from $1.70\,\text{TJ}$ to $0.88\,\text{TJ}$. For the planet, $GM = 4.0 \times 10^{14}\,\text{N m}^2\,\text{kg}^{-1}$. Resistive forces on the rocket may be taken as negligible. Use the expression in (b)(i) to find the distance from A to B.