Orbits and escape speed
Gravity as the centripetal force: orbital speed, period, the energy of a satellite, and how fast to leave for good.
The International Space Station moves at 7.7 km every second and never hits the ground. Is it escaping gravity, or falling?
A satellite moves in a circular orbit of radius r around a planet of mass M. Its orbital speed is…
A satellite is moved from an orbit of radius r to an orbit of radius 4r around the same planet. Its orbital speed becomes…
Satellite A orbits a planet at radius r with period T. Satellite B orbits the same planet at radius 4r. The period of B is…
A satellite of mass m moves in a circular orbit of radius r around a planet of mass M. Its total energy is…
The escape speed from the surface of a planet of mass M and radius R is…
A 500 kg research satellite is in low Earth orbit at radius 6.78 × 10⁶ m. Its kinetic energy is about…
The escape speed from a moon is 12 km s⁻¹. A probe moves in a circular orbit just above its surface. Its orbital speed is about…
An astronaut on the ISS says: “There is no gravity up here, that is why I float.” Use the ideas of this lesson to explain what is wrong.
Gravitational potential V_g = −GM / r is…
The distance to a mass is doubled. The field strength g becomes…
Key points
- • Gravity provides the centripetal force: v = √(GM / r), independent of the satellite’s mass.
- • T² = 4π² r³ / (GM), so r³ ∝ T² (Kepler’s third law).
- • KE = GMm / 2r, GPE = −GMm / r, TE = −GMm / 2r = −KE.
- • Escape speed v = √(2GM / R) makes the total energy zero. It is √2 times the orbital speed at that radius.
Formulas
- v = √(GM / r)
- T² = 4π² r³ / (GM)
- TE = −GMm / 2r
- vesc = √(2GM / R)
Key terms
- Orbital radius :
- distance from the centre of the planet to the satellite
- Escape speed :
- the minimum launch speed to reach infinity with no more propulsion
- Free fall :
- motion under gravity alone, which feels weightless
Common errors
- • Using the height above the surface as the orbital radius.
- • Thinking a faster satellite must be in a higher orbit.
- • Giving a bound satellite a positive total energy.