A stone, moving with speed $v$, follows a circular orbit of radius $r$ around the planet, as shown in Fig. 1.1. Show that the speed $v$ is given by the expression $v = \sqrt{\frac{GM}{r}}$, where $G$ is the gravitational constant. Explain your working.
A second stone starts from rest at infinity and moves towards the planet, as shown in Fig. 1.2. The stone does not strike the surface of the planet. Determine, in terms of the gravitational constant $G$ and the mass $M$ of the planet, the speed $v_0$ of the stone when it is a distance $x$ from the centre of the planet. Explain your working. You may assume that the stone is acted on only by the planet’s gravitational attraction.
Use your answer in (i) together with the expression in (a) to explain whether this stone could enter a circular orbit around the planet.