Define the gravitational potential at a point.
Starting from the expression for the gravitational potential due to a point mass, show that the gravitational potential energy $E_p$ of a point mass $m$ at distance $r$ from another point mass $M$ is $E_p = -\frac{GMm}{r}$, where $G$ is the gravitational constant.
Calculate the magnitude of the change in the gravitational potential energy $\Delta E_P$ of the comet as it travels from position $X$ to position $Y$.
State, with a reason, whether the change in gravitational potential energy in (b)(i) is an increase or a decrease.
Use your answer in (b)(i) to determine the speed, in $\text{km s}^{-1}$, of the comet at point $Y$.
A second comet crosses point $X$ with the same speed as the comet in (b) and in the same direction. This comet is slowly losing mass. Its mass at point $X$ is the same as that of the comet in (b). Suggest, with a reason, how the path of the second comet compares with the path shown in Fig. 1.1.