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and a surface temperature of 
. A probe with a mass of 
moves in a straight line towards the Sun. When it is at a distance 
from the centre of the Sun, the probe measures the gravitational field strength 
due to the Sun and the radiant flux intensity 
of radiation from the Sun. For the position of the probe where 
:
of the probe.
concentrated at its centre. A satellite of mass 
orbits the Earth such that the radius of the circular orbit is 
. For this satellite, write down expressions, in terms of 
, 
from the centre of Artemis of the gravitational potential 
is shown in the figure below
.
and radius 
. The planet may be considered to have all
. Show that, for the rocket to travel from a height 
above the planet’s surface, the change 
in the magnitude of the gravitational potential energy of the rocket is given by the expression
in a circular orbit about the Earth as shown in
and the mass of the Earth, considered to be a point mass at its centre, is 
.
.
. The gravitational field strength at the surface of the planet is 
. The planet may be assumed to be isolated in space and to have its mass concentrated at its centre.
. For point Y above the surface of the planet, the gravitational potential is 
. State, with a reason, whether point X or point Y is nearer to the planet.
and radius 
. The planet may be assumed to be isolated in space with its mass concentrated at its centre. A satellite of mass 
is in a circular orbit about the planet at a height 
above its surface. Calculate the gravitational potential energy of the satellite.
with its mass concentrated at its centre. The variation of the gravitational potential 
with distance x from the centre of the Earth is shown in the figure below
. By considering the gravitational potential at the Earth’s surface, determine a value for the mass of the Earth.
with all of its mass 
concentrated at its centre. Point A is a distance 
from the surface. Show that the difference in gravitational potential 
between points A and B is given by the expression
is the gravitational constant.
at distance 
from point mass 
is given by the expression
is the gravitational constant. Explain the significance of the negative sign in this expression.
is moving near to, and normal to, the surface of the planet. The mass moves away from the planet through a short distance 
. State and explain why the change in gravitational potential energy 
of the mass is given by the expression.
is the acceleration of free fall.
, as illustrated in the figure below.
at point P is shown in the figure below.
moves along the line joining the centres of the two stars, from star A towards star B. Use data from the figure above to calculate the change in kinetic energy of the rock when it moves from the point where 
to the point where 
. State whether this change is an increase or a decrease.
. Its mass of 
may be assumed to be a point mass at the centre of the planet.
is raised from the surface of the planet through a vertical height of
. Use the value of field strength given in (a) to determine the change in gravitational potential energy of the stone. Explain your working.
. The radius of the planet is 
. One possible path of the spacecraft as it approaches the planet is shown in the figure below.
. At point B, a distance of 
from the centre of the planet, the spacecraft has a speed of 
. The mass of the spacecraft is 
. For the spacecraft moving from point B to point A, show that the change in gravitational potential energy of the spacecraft is 
.
with its mass of 
concentrated at its centre. A rock of mass 
is situated on the surface of the Moon. Show that the change in gravitational potential energy of the rock in moving it from the Moon’s surface to infinity is 
.
may be assumed to have its mass 
concentrated at its centre. The magnitude of the gravitational potential at the surface of the sphere is 
. On the figure below, show the variation of the gravitational potential with distance 
from the centre of the sphere for values of 
to 
.
and mass 
. The planet has no atmosphere. A rock of mass 
moves directly towards the planet. Its distance from the centre of the planet changes from 
to 
. Calculate the change in gravitational potential energy of the rock.
with its mass of 
concentrated at its centre. A rock of mass 
rests on the surface of Mars. For this rock,
.
of a point mass 
at a distance 
from another point mass 
is given by
is the gravitational constant.
as it passes around a star of mass 
.
from the centre of the star and is moving at a speed of 
. At point Y, the comet passes its point of closest approach to the star. At this point, the comet is a distance of 
from the centre of the star. Both the comet and the star can be considered as point masses at their centres.
of the comet as it moves from position X to position Y.
