Escape velocity and energy in orbits

Curriculum links

Quoted word for word from the official documents.

Step by step

The caption is what the class reads on screen; the talking points are in your notes drawer (N) when you present. Steps marked Pupils only appear only in the pupil lesson.

1. 01

   Escape speed from energy

   Presented and for pupils

   Caption

   To escape, an object needs enough kinetic energy to climb right out of the gravitational field: ½mv² = GMm/R, so v = √(2GM/R).

   Talking points

   • The gravitational potential is V = −GM/R at the surface and zero at infinity, so escaping takes GM/R of work per kilogram. The object's mass m cancels, so escape speed is the same for a spacecraft and for a single atom.
   • JPL defines escape velocity as "the minimum velocity required for an object to escape the gravitational influence of the planet." NASA's fact sheet notes give it at the surface (the 1 bar level for the gas giants), "ignoring atmospheric drag".
   • NASA's Earth Fact Sheet gives 11.186 km/s.

   Ask the class

   Using GM = 3.986 × 10¹⁴ m³ s⁻² and Earth's mean radius R = 6.371 × 10⁶ m, calculate the escape speed from Earth's surface in km/s.

   Answer11.19 km/s. Answers from 11.14 to 11.24 are marked right.

   Whyv = √(2GM/R) = √(2 × 3.986 × 10¹⁴ ÷ 6.371 × 10⁶) = √(1.2513 × 10⁸) = 11,186 m/s = 11.19 km/s, as NASA gives. √(GM/R) = 7.91 km/s is a different speed: that of a circular orbit just above the surface.

   Sources 1 7 8 15Present from this step →

2. 02

   Escape from the Moon

   Presented and for pupils

   Caption

   The Moon has about an eightieth of Earth's mass and just over a quarter of its radius, so its escape speed is far lower.

   Talking points

   • NASA's Moon Fact Sheet gives GM = 0.00490 × 10⁶ km³ s⁻² (4.90 × 10¹² m³ s⁻²), a mean radius of 1,737.4 km (0.2727 of Earth's), a mass 0.0123 of Earth's and an escape velocity of 2.38 km/s.
   • JPL's more precise GM for the Moon, 4,902.800 km³ s⁻², gives 2.376 km/s, the same to three figures.

   Ask the class

   With GM = 4.90 × 10¹² m³ s⁻² and R = 1.737 × 10⁶ m, what is the Moon's escape speed in km/s?

   Answer2.38 km/s. Answers from 2.35 to 2.41 are marked right.

   Whyv = √(2GM/R) = √(2 × 4.90 × 10¹² ÷ 1.737 × 10⁶) = √(5.642 × 10⁶) = 2,375 m/s = 2.38 km/s, NASA's figure.

   Sources 2 9Present from this step →

3. 03

   Earth, the Moon, Mars and Jupiter

   Presented and for pupils

   Caption

   Escape speed grows with √(M/R). Jupiter, with 318 times Earth's mass, has the highest escape speed of any planet.

   Talking points

   • Mars: NASA's GM = 4.2828 × 10¹³ m³ s⁻² and mean radius 3,389.5 km give 5.03 km/s. Jupiter: GM = 1.26687 × 10¹⁷ m³ s⁻² and the equatorial radius at the 1 bar level, 71,492 km, give 59.5 km/s. Both match NASA's fact sheets.
   • Jupiter has no solid surface, so the radius used is a choice: JPL works from the mean radius, 69,911 km, and lists 60.20 km/s.
   • NASA's Planetary Fact Sheet gives the planets' escape speeds as Mercury 4.3, Venus 10.4, Earth 11.2, Mars 5.0, Jupiter 59.5, Saturn 35.5, Uranus 21.3 and Neptune 23.5 km/s.

   Ask the class

   Put these in order of escape speed, lowest first.

   They start in this order: Earth, Jupiter, The Moon, Mars.

   Answer

   1. The Moon
   2. Mars
   3. Earth
   4. Jupiter

   WhyThe Moon 2.38, Mars 5.03, Earth 11.19 and Jupiter 59.5 km/s (NASA).

   Sources 3 4 6 8Present from this step →

4. 04

   Energy in a circular orbit

   Presented and for pupils

   Caption

   In a circular orbit the kinetic energy is GMm/2r and the potential energy is −GMm/r, so the total energy is −GMm/2r. It is negative: the satellite is bound to the Earth.

   Talking points

   • From GMm/r² = mv²/r, ½mv² = GMm/2r. Adding the potential energy −GMm/r gives E = −GMm/2r.
   • To escape, the total energy must rise to zero, so the energy needed is GMm/2r, the same as the kinetic energy the satellite already has.
   • NASA's trajectory file of 7 October 2026 uses a mass of 470,200 kg for the ISS. At r = 6.80 × 10⁶ m that gives a kinetic energy of 1.38 × 10¹³ J and a total energy of −1.38 × 10¹³ J.
   • NASA's Space Station Facts and Figures page gives a lower mass, 419,725 kg, so the station's energy depends on which figure is used. The energy per kilogram, in the question, does not.

   Ask the class

   For the ISS at r = 6.80 × 10⁶ m, with GM = 3.986 × 10¹⁴ m³ s⁻², how much energy does each kilogram need to escape from its orbit? Give MJ per kg.

   Answer29.3 MJ/kg. Answers from 29 to 29.6 are marked right.

   WhyPer kilogram the energy needed is GM/2r = 3.986 × 10¹⁴ ÷ (2 × 6.80 × 10⁶) = 2.93 × 10⁷ J/kg = 29.3 MJ/kg. GM/r = 58.6 MJ/kg is too much: it leaves out the kinetic energy the station already has.

   Sources 1 11 12Present from this step →

5. 05

   Escape speed and orbital speed

   Presented and for pupils

   Caption

   At any distance r, the escape speed √(2GM/r) is √2 times the speed of a circular orbit there, √(GM/r).

   Talking points

   • For the ISS at r = 6.80 × 10⁶ m: 7.66 km/s in orbit, and 7.66 × √2 = 10.83 km/s to escape from that height.
   • The same factor links the two speeds at the surface: 7.91 km/s for an orbit just above the ground, 11.19 km/s to escape.

   Ask the class

   At the same distance from a planet, how does the escape speed compare with the speed of a circular orbit?

   1. AThey are the same
   2. BThe escape speed is √2 times as fastRight answer
   3. CThe escape speed is twice as fast
   4. DThe escape speed is four times as fast

   Why√(2GM/r) ÷ √(GM/r) = √2 = 1.414. For the ISS, 7.66 km/s × 1.414 = 10.83 km/s.

   Sources 1 11Present from this step →

6. 06

   Why the Moon has so little air

   Presented and for pupils

   Caption

   The Moon's atmosphere is almost nothing. NASA gives two main reasons Earth holds far more: its stronger gravity, and its more active sources of gas.

   Talking points

   • NASA: the Moon's atmosphere "contains about one million billion (10¹⁵) times fewer molecules per cubic centimeter than Earth’s does. This is primarily because Earth is more massive (so it has a stronger gravitational pull holding its atmosphere in place) and Earth has more active sources of atmospheric gases (like erupting volcanoes). Our planet’s magnetic field also helps to protect and preserve our atmosphere."
   • NASA's fact sheets put the total mass of the Moon's atmosphere at about 25,000 kg and Earth's at 5.1 × 10¹⁸ kg.
   • NASA: the lunar atmosphere is almost entirely helium, neon and argon. Some of its particles settle to the ground or escape into space while new impacts send others up.

   Ask the class

   Which of these does NASA give as reasons Earth holds far more atmosphere than the Moon? Choose all that are.

   More than one answer is right.

   1. AEarth is more massive, so its gravity holds its atmosphere in placeRight answer
   2. BEarth has more active sources of gas, such as erupting volcanoesRight answer
   3. CEarth's magnetic field helps to protect and preserve its atmosphereRight answer
   4. DEarth is much closer to the Sun than the Moon is

   WhyNASA names all three. The Moon goes round the Sun with the Earth, and NASA's fact sheets give both the same solar irradiance, 1361.0 W/m², so in effect they are the same distance from the Sun.

   Sources 1 2 10Present from this step →

7. 07

   How fast is helium on the Moon?

   Presented and for pupils

   Caption

   On the Moon's hot day side, helium atoms have an r.m.s. speed of about two thirds of the Moon's escape speed.

   Talking points

   • The kinetic theory of gases gives ½m(c_rms)² = (3/2)kT, so c_rms = √(3kT/m) = √(3RT/M), with M the molar mass.
   • NASA's Moon Fact Sheet gives the equatorial temperature range as 95 K to 390 K. The IUPAC commission (CIAAW) gives helium's standard atomic weight as 4.002602, so its molar mass is 4.003 × 10⁻³ kg/mol. CODATA gives R = 8.314 J mol⁻¹ K⁻¹.
   • Atoms in a gas have a spread of speeds about the r.m.s. value, the Maxwell-Boltzmann distribution in OCR's specification. The closer c_rms comes to the escape speed, the larger the share of atoms fast enough to leave. OCR's guidance for escape velocity includes "Predicting the escape velocity of atoms from the atmosphere of planets."

   Ask the class

   Using c_rms = √(3RT/M) with R = 8.314 J mol⁻¹ K⁻¹, T = 390 K and M = 4.003 × 10⁻³ kg/mol, find the r.m.s. speed of helium atoms in km/s.

   Answer1.56 km/s. Answers from 1.54 to 1.58 are marked right.

   Whyc_rms = √(3 × 8.314 × 390 ÷ 4.003 × 10⁻³) = √(2.430 × 10⁶) = 1,559 m/s = 1.56 km/s, 0.66 of the Moon's escape speed of 2.38 km/s.

   Sources 2 13 14 15 16Present from this step →

8. 08

   Find the lowest escape speed

   Pupils only

   Caption

   Your turn. Of the eight planets, which has the lowest escape speed? Use the arrows to go to it.

   Talking points

   • NASA's Planetary Fact Sheet gives Mercury 4.3 km/s and Mars 5.0 km/s, the two lowest. JPL lists 4.25 km/s for Mercury.
   • From NASA's Mercury Fact Sheet, GM = 0.022032 × 10⁶ km³ s⁻² and the mean radius, 2,439.7 km, give √(2GM/R) = 4.25 km/s.

   Pupil task

   Use the arrows to step from planet to planet, and stop on the planet with the lowest escape speed.

   HintWork out √(2GM/R) for the two smallest planets. Mars: GM = 4.28 × 10¹³ m³ s⁻², R = 3.39 × 10⁶ m. Mercury: GM = 2.20 × 10¹³ m³ s⁻², R = 2.44 × 10⁶ m.

   The viewer checks the task as the pupil works and says when it is done.

   Sources 3 5 6 8Open this step as a pupil →

Sources

Every fact in this lesson comes from these sources, and each step lists the ones it uses.

1. Earth Fact Sheet, NASA NSSDCA
2. Moon Fact Sheet, NASA NSSDCA
3. Mars Fact Sheet, NASA NSSDCA
4. Jupiter Fact Sheet, NASA NSSDCA
5. Mercury Fact Sheet, NASA NSSDCA
6. Planetary Fact Sheet (metric), NASA NSSDCA
7. Notes on the Planetary Fact Sheets, NASA NSSDCA
8. Planetary Physical Parameters, JPL Solar System Dynamics
9. Astrodynamic Parameters, JPL Solar System Dynamics
10. Lunar Atmosphere, NASA Science
11. ISS trajectory data (Orbital Ephemeris Message from NASA Johnson Space Center), linked from this NASA page
12. Space Station Facts and Figures, NASA
13. Molar gas constant, CODATA 2022, NIST
14. Helium: standard atomic weight, IUPAC Commission on Isotopic Abundances and Atomic Weights
15. AQA AS and A-level Physics specification (7407, 7408), version 1.3
16. OCR A Level Physics A (H556) specification, version 3.0