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Lesson 26 · Falling around the Earth
An orbit is a fall that misses
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An orbit is a fall that misses

A satellite is not beyond gravity; it is falling continuously and moving sideways fast enough to keep missing. Set gravity's pull equal to the inward force a circle demands — GMmr2=mv2r\dfrac{GMm}{r^2} = \dfrac{mv^2}{r} — cancel the satellite's own mass mm from both sides, and one line of algebra gives v=GMrv = \sqrt{\dfrac{GM}{r}}: read aloud, v equals the square root of big G M over r.

The cast: vv is the orbital speed in m/s, MM is the central mass in kilograms — the planet, not the satellite — and rr is the orbital radius in metres, measured from the planet's CENTRE. That last word is where marks die. An orbit "400 km up" has r=6370+400=6770r = 6370 + 400 = 6770 km, not 400: the altitude is a passenger, the radius is the quantity. Note too what is absent — the satellite's own mass cancelled, so a bolt and a space station on the same orbit fly at exactly the same speed. And push the units through: m3/(kg⋅s2)×kg÷m\mathrm{m^3/(kg \cdot s^2)} \times \mathrm{kg} \div \mathrm{m} leaves m2/s2\mathrm{m^2/s^2}, whose root is m/s. If your rearrangement cannot do that, it is wrong, no appeal — though units that do work out never prove you right.

Mrv

v=GMrv = \sqrt{\frac{GM}{r}}

  • vv= Orbital velocity (speed)
  • MM= Central mass (mass)
  • rr= Orbital radius (length)
Orbital Velocity solver →
rmvFc

Fc=mv2rF_c = \frac{m v^2}{r}

  • FcF_c= Centripetal force (force)
  • mm= Mass (mass)
  • vv= Speed (speed)
  • rr= Radius (length)
Centripetal Force (F = mv²/r) solver →