13.2 Gravitational Potential

The Earth may be assumed to be a uniform sphere of radius R = 6.38 × 10⁶ m, with its mass M concentrated at its centre.

A satellite of mass m orbits the Earth at a height h₁ above the Earth's surface.

The satellite moves to a distance h₂ away from the surface of the Earth.  

Derive an expression for the change in gravitational potential energy as the satellite moves from h₁ to h₂.

When the satellite reaches a height h₂ above the surface of the Earth, it begins to drift out of its orbit. 

Determine the distance moved by the satellite once it has drifted to the point the gravitational potential energy has increased by 20% of its initial value at h₁. 

Assume that the satellite moves along the same line connecting points h₁, h₂ and the centre of the Earth. 

A space shuttle of mass 3 × 10⁶ kg is travelling from the moon back to Earth. It accelerates uniformly from launch at 2.3 m s⁻². It has enough propellant to provide thrust for the first 155 seconds. 

The mass of the moon is 7.35 × 10²² kg and the mean radius is 1740 km. 

The Moon has a radius of approximately 27% that of the Earth, and a mass of 1.2% that of the Earth. 

The gravitational potential at a point A about the surface of a planet is –7.2 MJ kg^–1. 

For a point B above the surface of the planet, the gravitational potential is –9.1 MJ kg^–1.

 State, with a reason, whether point A or B is further away from the planet.

A rocket is landing onto the surface of Mercury.

Table 1.1 gives data for the speed of the rocket at two heights about the surface of Mercury, after the rocket engine has been switched off.

Table 1.1

Mercury may be assumed to be a uniform sphere of radius R, with its mass M concentrated at its centre. The rocket, after the engine has been switched off, has mass m. The change in gravitational potential energy of the rocket between the two heights is ΔE_p.

Show that the mass M of Mercury is given by the expression

Write an expression for the chang in kinetic energy of the rocket.

Venus may be assumed to be an isolated sphere of radius 6100 km with its mass, M, at 4.9 × 10²⁴ kg concentrated at its centre. 

An object is projected vertically from the surface of Venus so that it reaches an altitude of 7.2 × 10⁴ km.

Explain why the equation MgΔh where g is the gravitational field strength of Venus and Δh is the height of the object, cannot be used to calculate the object's gravitational potential energy.

The change in the gravitational potential of a different object is 4.2 MJ kg^–1. This object ends up in orbit around Venus.

Calculate the radius of orbit of this object.

Calculate the change in work done per unit mass of the object projected vertically in part (a).

Show that the initial speed of the object in (a) is 1.1 km s^–1, assuming air resistance is negligible.