Two point masses are isolated in space and are separated by a distance $x$. State an equation that links the gravitational force $F$ between the two masses with the masses $M$ and $m$. State the name of any other symbol used.
A spacecraft is to be placed in a circular orbit around a spherical planet. The planet may be taken to be isolated in space. The mass of the planet, assumed to be concentrated at its centre, is $7.5 \times 10^{23}\,\text{kg}$. The radius of the planet is $3.4 \times 10^{6}\,\text{m}$.
The spacecraft is to orbit the planet at a height of $2.4 \times 10^{5}\,\text{m}$ above the planet’s surface. At this altitude, there is no atmosphere. Show that the speed of the spacecraft in its orbit is $3.7 \times 10^{3}\,\text{m s}^{-1}$.
A possible route taken by the spacecraft as it nears the planet is shown in Fig. 1.1. The spacecraft enters the orbit at point A with speed $3.7 \times 10^{3}\ \text{m s}^{-1}$. At point B, which is $5.00 \times 10^{7}\ \text{m}$ from the planet’s centre, the spacecraft’s speed is $4.1 \times 10^{3}\ \text{m s}^{-1}$. The spacecraft has mass $650\ \text{kg}$. For the spacecraft travelling from point B to point A, show that the change in gravitational potential energy of the spacecraft is $8.3 \times 10^{9}\ \text{J}$.
By considering the changes in gravitational potential energy and in kinetic energy of the spacecraft, determine whether the total energy of the spacecraft increases or decreases as it moves from point B to point A. No numerical answer is required.