Orbital Velocity Derivation

For circular orbit, gravity provides centripetal force:

mv_{0}^{2}/r = GMm/$r^{2}$ v_{0} = √(GM/r) = √($gR^{2}$/r)

Near surface (r ≈ R): v_{0} = √(gR) ≈ 7.9 km/s

Orbital Period

T = 2πr/v_{0} = 2π√($r^{3}$/GM) = (2π/R)√($r^{3}$/g) × (1/√R)

Near surface: T = 2π√(R/g) ≈ 84.6 min

Effect of Raising Orbit

The paradox: adding energy to a satellite makes it go to a higher orbit where it moves slower.

Centripetal Force Requirement

Gravity is the centripetal force — there is no separate centripetal force. The satellite is in a state of continuous free fall toward Earth. Because it has horizontal velocity, it keeps "falling around" Earth.

Geostationary Orbit Derivation

From Kepler's Third Law with T = 24 h:

$r^{3}$ = G$MT^{2}$/(4π^{2}) = gR^{2}$$T^{2}/(4π^{2})

r = (gR^{2}$$T^{2}/4π^{2})^(1/3) = (10 × ($6.4 \times 10^{6}$)^{2} × (86400)^{2}/(4π^{2}))^(1/3)

r ≈ 42,164 km from Earth's centre Height above surface ≈ 35,786 km

Properties:

• Equatorial plane (inclination = 0°)
• West-to-east direction (same as Earth's rotation)
• Appears stationary from ground
• Orbital speed ≈ 3.1 km/s