Syllabus Edition

First teaching 2023

First exams 2025

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Gravitational Potential (CIE A Level Physics)

Exam Questions

1 hour9 questions
1a
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1 mark

State the physical quantity defined as the energy possessed by an object due to its position in a gravitational field.

1b
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2 marks

Define gravitational potential at a point.

1c
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5 marks

Gravitational potential, φ, is also defined by the equation

   phi space equals space minus fraction numerator G M over denominator r end fraction
(i)
Identify each quantity in the equation.
[3]
(ii)
State the significance of the negative sign in the equation.
[2]
1d
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1 mark

Fig. 1.1 shows a person on the surface of the Earth, a ball falling towards the centre of the Earth and a satellite in orbit around the Earth. 

13-2-1d-e-original-image-name-5-1-4-weight-force
Fig. 1.1
 

Identify the object which experiences the largest gravitational potential. 

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2a
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1 mark

Place a tick () next to the equation in Table 1.1 which represents the gravitational potential energy of two point masses m and M separated by a distance r. 

 
Table 1.1
 
Equation  
phi space equals space minus space fraction numerator G M over denominator r end fraction  
E subscript p space equals space minus space fraction numerator G M m over denominator r end fraction  
E subscript p space equals space m g h  
g space equals space fraction numerator G M over denominator r squared end fraction  

2b
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1 mark

The diagram in Fig. 1.2 shows two satellites m1 and m2 in orbit around the moon. 

  13-2-2b-e-original-image-name-change-in-gpe-and-new-image-here-m1-m2-orbit-moon-sqe-cie-a-level

Fig. 1.2
  

Satellite m1 is in orbit at a distance of r1 from the centre of the moon and satellite m2 is in orbit at a distance of r2 from the centre of the moon. 

 

State which of the satellites has the higher gravitational potential energy. 

2c
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2 marks

Satellite m1 has a mass of 300 kg and orbits above the centre of the moon at a radius r1 of 500 m.

The mass of the moon is 7 × 1022 kg.

Calculate the gravitational potential energy of the satellite in this position. Newton's gravitational constant, = 6.67 × 10−11 N m2 kg−2

2d
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3 marks

State why the equation for gravitational potential energy in (a) has a negative sign and why your answer to (c) is negative.

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3a
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2 marks

Explain why gravitational potential is always negative.

3b
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3 marks

Fig. 1.1 shows a satellite orbiting Mars, M. The satellite is moves from orbit X to orbit Y.

 
13-2-3b-e-orbits-grav-pot-esq-cie-a-level
Fig. 1.1
 

The gravitational potential due to Mars in each of these orbits is:

 
Orbit X: −3.56 MJ kg−1
Orbit Y: −1.23 MJ kg−1
 

Calculate the gravitational potential difference as the satellite moves from orbit X to orbit Y.

3c
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4 marks

The gravitational potential energy at a particular point is given by the equation:

       E subscript p space equals negative fraction numerator G M m over denominator r end fraction 

Define each of the terms in the equation.

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1a
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2 marks

Mars is considered to be an isolated sphere of radius R with its mass concentrated at its centre. 

The variation of the gravitational potential φ with distance x from the centre of Mars is shown in Fig 1.1.

 

13-2-1a-m-mars-gravitational-potential-graph

Fig 1.1
 

Explain why the values for φ in Fig 1.1. are negative.

1b
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3 marks

The radius R of Mars is 3400 km.

By considering the gravitational potential at Mars' surface, determine a value for its mass.
1c
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4 marks

A meteorite is at rest at infinity. The meteorite travels from infinity towards Mars. 

Calculate the speed of the meteorite when it is at a distance of 3R above Mars' surface. Explain your working.

1d
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3 marks

Mars has two moons, Phobos and Deimos, as shown in Fig 1.2.

13-2-1d-m-mars-and-its-moons
Fig 1.2
 

Calculate the work done to move a satellite of 210 kg from the orbit of Deimos to the orbit of Phobos.

 

Orbital radius of Phobos = 9400 km

Orbital radius of Deimos = 2.3 × 104 km

Work done = ............................. MJ 

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2a
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2 marks

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.

2b
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3 marks

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

height / km speed / m s–1
h1 = 15 × 103 v1 = 7215 
h2 = 3.7 × 103 v2 = 7020

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 ΔEp.

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

M space equals fraction numerator increment E subscript p over denominator G m open parentheses fraction numerator 1 over denominator R plus h subscript 1 end fraction space minus fraction numerator space 1 over denominator R plus h subscript 2 end fraction close parentheses space end fraction

2c
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1 mark

Write an expression, in terms of m, v1 and v2 for the change in kinetic energy of the rocket.

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3a
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1 mark

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.

3b
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3 marks

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.

3c
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2 marks

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

 
Change in work done per unit mass = .................... MJ kg–1
3d
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3 marks

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

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1a
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3 marks

A spherical planet has a mass and radius R. The planet may be considered to have all its mass concentrated at its centre.

A rocket, of mass m is launched from the surface of the planet.

The rocket engines are stopped when the rocket is at a height above the surface of the planet, as shown in Fig 1.1.

13-2-1a-h-planet-rocket-gpe-sq-cie-a-levelFig. 1.1

The change in gravitational potential energy of the rocket as it travels from a height to a height 4R above the planet's surface is equal to   

 

      increment E subscript p space equals space x open parentheses fraction numerator G M m over denominator R end fraction close parentheses 

Determine the value of x

1b
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4 marks

At a distance away from the surface of the planet the rocket has a velocity 2and at a distance of 4away from the planet a velocity v

Show that the expression for the thermal energy lost increment Q to the atmosphere of the planet as the rocket moves from to 4R is given by

increment Q space equals space fraction numerator 3 m over denominator 2 end fraction open parentheses v squared space minus space fraction numerator G M over denominator 5 R end fraction close parentheses

1c
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4 marks

The rocket travels towards a moon orbiting the planet

The distance from the centre of the planet to the centre of the moon is 200R. The moon has a mass M over 16 where M is the mass of the planet.  

Determine the distance, in terms of R, from the centre of the planet at which 

         fraction numerator increment E subscript P over denominator increment r end fraction space equals space 0 

Give your answer to the nearest whole number.

1d
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3 marks

On the axes of Fig. 1.2, sketch a graph to show the variation of the gravitational potential with distance along a line between the surface of the planet and the surface of the moon.

 

13-2-1d-h-13-2-h-potential-distance-graph-cie-ial-sq

Fig. 1.2

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2a
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4 marks

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 orbits the Earth at a height h1 above the Earth's surface.

Derive an expression for the total energy of the satellite in terms of G, M, m, h1 and R.
 
2b
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2 marks

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.

2c
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4 marks

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. 

Mass of the Earth = 5.97 × 1024 kg
Mass of the satellite = 1720 kg
Height h1 = 2.23 × 106 m
Height h2 = 3.05 × 106 m

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3a
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3 marks

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. 

Calculate the work done by the rocket during the first 155 seconds after launch. State any assumptions you have made.
3b
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3 marks

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

Show that the gravitational potential at the surface of the Earth is about 24 times greater than the gravitational potential at the surface of the moon.

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