Orbital Speed, Kepler’s Third Law and Escape Velocity: Derivations and Worked Examples (IB Physics)
The International Space Station moves at about 7.7 km every second and never hits the ground. It is not escaping gravity: it is falling around the Earth. This guide derives the orbital speed, the period, the energy of a satellite and the escape speed from one idea, that gravity is the centripetal force.
Gravity is the centripetal force
A satellite of mass m moves at constant speed v in a circle of radius r around a planet of mass M, where r is measured from the centre. Gravity is the only force, so it provides the centripetal force: GMm ÷ r² = mv² ÷ r. Cancel m and one factor of r: v² = GM ÷ r, so v = √(GM ÷ r).
Notice that the satellite’s mass cancels, so heavy and light satellites move at the same speed at the same height, and a higher orbit means a slower satellite.
The period and Kepler’s third law
One orbit is a distance 2πr, so v = 2πr ÷ T. Put it into v² = GM ÷ r and rearrange: T² = 4π²r³ ÷ GM. Since 4π² ÷ GM is a constant for a given planet, T² is proportional to r³, which is Kepler’s third law. A satellite at 4 times the radius takes 8 times as long to orbit.
Worked example: the International Space Station
The ISS orbits at r = 6.78 × 10⁶ m from the centre of the Earth, about 400 km above the surface. v = √(GM ÷ r) = √(6.67 × 10⁻¹¹ × 5.97 × 10²⁴ ÷ 6.78 × 10⁶) = 7.66 × 10³ m/s. The period is T = 2πr ÷ v = 5.56 × 10³ s, about 93 minutes, which is roughly 15 orbits a day.
Why astronauts float
At the ISS orbit the field strength is g = GM ÷ r² = 8.7 N/kg, about 88% of its value at the surface. Gravity has not gone away: it is what keeps the station in orbit. The station and the astronauts are in free fall together, so nothing pushes on them. That is the feeling of weightlessness.
The energy of a satellite
In a circular orbit the kinetic energy is ½mv² = GMm ÷ 2r. The gravitational potential energy is −GMm ÷ r, negative because gravity only attracts. The total energy is the sum:
- •KE = +GMm ÷ 2r
- •GPE = −GMm ÷ r
- •Total = −GMm ÷ 2r, which is minus the kinetic energy.
- •Example: a 500 kg satellite at r = 6.78 × 10⁶ m has KE = +1.47 × 10¹⁰ J, GPE = −2.94 × 10¹⁰ J and total energy −1.47 × 10¹⁰ J. A negative total energy means the satellite is bound.
Escape velocity
The escape speed is the minimum launch speed that lets an object, with no further propulsion, get infinitely far away. Its total energy must reach zero: ½mv² − GMm ÷ R = 0, so v_esc = √(2GM ÷ R). The mass cancels again.
For the Earth’s surface, v_esc = 1.12 × 10⁴ m/s, about 11 km/s. That is √2 times the circular orbital speed at the same radius (7.9 km/s just above the surface): twice the kinetic energy gives √2 times the speed.
Common mistakes
- •Using the height above the surface as the orbital radius. The radius is measured from the planet’s centre.
- •Thinking a faster satellite must be in a higher orbit. Higher orbits are slower.
- •Giving a bound satellite a positive total energy.
- •Saying there is no gravity on the space station.
Frequently asked questions
Set gravity equal to the centripetal force: GMm ÷ r² = mv² ÷ r. Cancel m and r once to get v² = GM ÷ r, so v = √(GM ÷ r).
v = √(2GM ÷ R) = 1.12 × 10⁴ m/s, about 11.2 km/s, using the Earth’s mass and radius. It does not depend on the mass of the object.
Its kinetic energy is +GMm ÷ 2r but its gravitational potential energy is −GMm ÷ r, so the total is −GMm ÷ 2r. Negative total energy means it is bound to the planet.
They are in free fall with the space station, both accelerating toward the Earth at the same rate, so there is no contact force pushing on them.