AQA A-level Physics 7408 · Paper 2 · Spec 3.7.1–3.7.2

Gravitational Fields

Free notes, interactive simulations, exam questions and a printable booklet.

Colours on this page: g field strength · V potential · F force · v velocity · E energy

What does a gravitational field look like?

Every mass is surrounded by a field. Drag the probe mass anywhere and watch the field strength, the potential and the force change. The arrows on the field lines get bigger where the field is stronger.

The field (drag the probe anywhere)
Distance r
g = GMr2
V = −GMr
F = mg
Ep = mV
Graphs of g and V against r

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Equipotentials are drawn at equal steps of potential, so they spread out where the field is weaker. Field lines always cross them at right angles. Arrows at the probe are drawn to scale. Planet sizes and the Earth–Moon distance are not to scale; the numbers are calculated with the real values.

Orbits and escape

A satellite is falling all the time. It moves sideways so fast that the Earth curves away beneath it. Watch the Earth turn and the satellite go round: every orbit here keeps time with the Earth’s rotation.

The orbit (seen from above the North Pole; orbits drawn in the equatorial plane; radii not to scale)
Speed
v = √GMr
T = 2πrv
g at the orbit
The energies
Ek = GMm2r
Ep = −GMmr
E = −GMm2r
The satellite’s own mass cancels out of v and T. Change m: only the energies change.
Kepler’s third law from real data: T2 against r3

Flying off the Earth. Spin the Earth faster and watch the forces on an object at the equator, until the ground stops pushing on it and it lifts off. This is worked example E1 in the booklet.

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Data: G = 6.67 × 10−11 N m2 kg−2, mass of the Earth 5.97 × 1024 kg, radius 6.37 × 106 m, one turn of the Earth (sidereal day) 86 164 s. The orbit view squeezes large radii so they fit, but every satellite moves at its true angular speed on the same clock as the Earth. Real orbits can be tilted (the ISS orbit is inclined at 51.6°); here every orbit is drawn in the equatorial plane. Non-circular orbits are not examined by AQA. Map outlines simplified from Natural Earth (public domain).

Key equations for gravitational fields

Newton’s law of gravitation
AQA formula sheet
F = Gm1m2r2
Force between point masses, N. r is measured centre to centre
Field strength
AQA formula sheet
g = Fm
g = GMr2
Force per unit mass, N kg−1. The second form is for a radial field
Potential
AQA formula sheet
V = −GMr
J kg−1, zero at infinity, negative everywhere else
Work and potential
AQA formula sheet
ΔW = mΔV
Also the change in potential energy. No work along an equipotential
Field from potential
AQA formula sheet
g = −ΔVΔr
Gradient of the V–r graph. Area under g–r = ΔV
Orbital speed
Useful relationship: derive it
v = √GMr
Circular orbit; the satellite’s mass cancels
Kepler’s third law
Useful relationship: derive it
T2 = 4π2r3GM
So T2 ∝ r3 for everything orbiting the same body
Energy in orbit
Useful relationship
E = −GMm2r
Ek = GMm/2r and Ep = −GMm/r
Escape velocity
Useful relationship: derive it
vesc = √2GMR
From ½mv2 = GMm/R; independent of the object’s mass

Quick catch

Three rounds of five multiple-choice questions, each round harder than the last. Pick A, B, C or D and the reasons appear straight away. Get 4 of 5 right to unlock the next round.

0 of 5 right

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Where it comes from

Three results that AQA expects you to understand, built up step by step. The first two can be asked for in the exam; the third shows where the potential comes from.

1
Gravity is the centripetal force
A satellite of mass m moves in a circle of radius r around a body of mass M. Gravity is the only force on it, so gravity provides the centripetal force.
GMmr2 = mv2r
2
The orbital speed
Cancel m and one r. The satellite’s own mass has gone.
v = √GMr
3
Bring in the period
In one period the satellite goes once round the circle, so v = 2πr/T. Square it and put it into step 1.
GMr = 4π2r2T2
4
Rearrange
4π2/GM is the same for everything orbiting the same body, so T2 ∝ r3.
T2 = 4π2GMr3
One orbit at a time: T2 against r3
1
What escaping means
An unpowered object escapes if it can reach infinity, where its potential energy is zero. At the escape speed it arrives there with no energy to spare: its kinetic energy has fallen to zero. In the simulation the object is launched sideways from the surface: below the escape speed it follows a closed ellipse and comes back round; at the escape speed its path opens out and it never returns.
2
Energy at the surface
At the Earth’s surface (radius RE) its potential energy is −GMm/RE. The total energy must be at least zero.
½mv2 − GMmRE = 0
3
Solve for the speed
The mass m cancels, so every object needs the same escape speed, and the direction of launch does not matter (ignoring air resistance).
vesc = √2GMRE
4
Compare with orbit speed
At the surface, orbit speed is √(GM/RE), so the escape speed is √2 times larger: 11.2 km s−1 against 7.91 km s−1 for the Earth.
Launch it and see: Beyond Contact

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Not examined by AQA. This uses integration from A-level Maths, to show where V = −GM/r comes from.
1
Work against the field
r is the distance from the centre of the Earth; above the surface r = RE + h. Moving 1 kg outwards a small distance dr against the field takes work g dr, with g = GM/r2.
2
Add it up from r to infinity
This is the area under the g–r graph from r to infinity.
W = ∫r∞ GMr2 dr = GMr
3
Potential is minus this
Potential is the work done bringing 1 kg in from infinity. The field does the work, so the work we do is negative.
V = −GMr
4
And back again
Differentiate: dV/dr = GM/r2. The field points the way V falls, so g = −dV/dr, the calculus form of g = −ΔV/Δr.
Area under the g–r graph, and V against r

Test yourself

35 exam-style questions in three levels: Core, Exam-style and Stretch for A*. Each level shows a few at a time. Press Try another question for a different one. Type numbers without units. For the explain parts, write your answer, then mark it against the mark scheme.

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Download the booklet

Notes, worked examples, equation cards, 19 quick checks and 26 exam-style questions on gravitational fields, to print and write on. The answers are at the back.

Gravitational Fields: notes and practice booklet

Print it, write in the spaces, then check your answers at the back.

  • PDF
  • 54 pages
  • 2.8 MB
  • Version 1.5
  • 14 October 2026
  • Free
Download the free Gravitational Fields booklet (PDF) (opens in a new tab)

It opens in a new tab. Use the download or print button in the PDF viewer to keep a copy. Print it double-sided and write in the spaces. In the PDF, each question links to its answer and back again.

Teachers: free for classroom use. Please share this page link rather than the file, so students always get the latest version.

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Gravitational fields: common questions

Why is gravitational potential always negative?

Potential is defined as zero at infinity, far from every mass. Gravity always attracts, so a mass brought in from infinity is pulled in: the field does the work, and the work we do is negative.

V = −GMr

Moving away from a planet, V rises towards zero. So going from −60 MJ kg−1 to −40 MJ kg−1 is an increase in potential of 20 MJ kg−1.

Exam trap. A definition needs both “work done per unit mass” and “from infinity”.

Do I use the height or the distance from the centre?

Always the distance from the centre. In F = Gm1m2/r2, g = GM/r2, V = −GM/r and every orbit equation, r is measured from the centre of the planet.

r = R + h

For the ISS, 420 km up: r = 6.37 × 106 + 4.20 × 105 = 6.79 × 106 m. Using 4.20 × 105 m gives an answer hundreds of times too big.

Exam trap. Convert km to m before you substitute, then add the radius.

Why does a satellite speed up when air resistance drops it to a lower orbit?

Air resistance removes energy, so the total energy E = −GMm/2r becomes more negative and the orbit shrinks. In a smaller orbit the speed is larger:

v = √GMr

The potential energy falls by twice as much as the kinetic energy rises. Air resistance takes away the difference. Try it with the “Atmospheric drag (low orbits only)” button above.

Are astronauts on the ISS weightless because there is no gravity?

No. At the height of the ISS the field strength is about 8.6 N kg−1, almost 90% of its value on the ground. The astronauts and the station are both in free fall, with the same acceleration towards the Earth.

Gravity is the only force on them, and it provides the centripetal force for the orbit. The floor does not push on the astronauts, and that push is what we feel as weight.

Exam trap. “There is no gravity in space” scores nothing.