Define gravitational potential at a point, in terms of work done per unit mass.
The Moon can be treated as an isolated uniform sphere with mass $7.3 \times 10^{22}\,\text{kg}$ and radius $1.7 \times 10^{6}\,\text{m}$. Calculate the gravitational potential at the surface of the Moon. Give a unit with your answer.
An isolated uniform spherical planet has gravitational potential $\phi$ at its surface. A particle of mass $m$ is launched vertically upwards from the surface. The particle is given just enough kinetic energy to travel an infinite distance away from the planet, escaping from the planet’s gravitational pull, without any extra work being done on it. Determine an expression, in terms of $m$ and $\phi$, for the gravitational potential energy $E_p$ of the particle at the surface of the planet.
Show that the upward launch speed $v$ from the surface of the planet is $v = \sqrt{-2\phi}$.
A particle is moving upwards at the Moon’s surface. Use your answer in (a)(ii) and the expression in (b)(ii) to determine the minimum speed that will allow the particle to escape from the Moon’s gravitational pull.
Hydrogen may be treated as an ideal gas. The mass of a hydrogen molecule is $3.34 \times 10^{-27}\,\text{kg}$. Calculate the root-mean-square (r.m.s.) speed of a hydrogen molecule in hydrogen gas at a temperature of $400\,\text{K}$.
The Moon’s surface can reach temperatures of about $400\,\text{K}$ when it is in direct sunlight. Use your answers in (c) and (d) to suggest a reason why the Moon does not have an atmosphere made up of hydrogen.