Satellite Speed: UNIZOR.COM - Physics4Teens - Mechanics - Gravity, Weight

Notes to a video lecture on http://www.unizor.com

Satellite Speed

Our task is to find out a linear speed V of a satellite that freely rotates around a planet of mass M on a circular orbit of radius R.

The gravity force, acting on a satellite of mass m and keeping it on a circular orbit with constant angular and linear speed, according to the Law of Universal Gravitation, equals to
F = G·M·m /R²
where G=6.67·10⁻¹¹(N·m²/kg²) is a Universal Gravitational Constant.

On the other hand, according to Rotational Kinematics, that same force gives a satellite a centripetal acceleration
a = V²/R

Applying the Newton's Second Law
F = m·a,
we obtain an equation that connects radius of an orbit, linear speed of a satellite and mass of a planet:
F = m·a = G·M·m /R²

Using the expression of centripetal acceleration above, this results in the following:
m·V² /R = G·M·m /R²

Notice that mass of a satellite m cancels out and the resulting expression for a satellite linear speed on an orbit is
V² = G·M /R
V = √G·M/R

The above formula allows to calculate the period T of rotation of a satellite - the time required to make a complete circle around a planet:
T = 2πR/V = 2π√R³/(G·M)