Orbital Velocity
v_orb = √(GM/r) — satellite at altitude h with period and g(h).
Key Notes
Orbital velocity v_orb is the speed required for an object to maintain a stable circular orbit around a central body.
Set centripetal force = gravity: mv²/r = GMm/r² ⇒ v_orb = √(GM/r).
At Earth's surface (r ≈ R): v_orb = √(GM/R) = √(gR) ≈ 7.9 km/s (first cosmic velocity).
Period: T = 2πr/v = 2π·√(r³/GM) — Kepler's third law.
Higher orbits ⇒ slower velocity but longer period.
Geosynchronous orbit (T = 24 hours): r ≈ 42,164 km, v ≈ 3.07 km/s.
Total energy in circular orbit: E = −GMm/(2r). Negative ⇒ bound.
Escape velocity = √2 × orbital velocity at the same r.
Formulas
Orbital velocity
Decreases with increasing r.
Orbital period (Kepler 3)
T² ∝ r³.
First cosmic velocity (low orbit)
Minimum speed to circle Earth just above surface.
Total energy
Half of PE, opposite sign of KE.
Kepler's third law (general)
Same form for all bodies orbiting same central mass.
Important Points
Faster orbits are CLOSER — counterintuitive but follows from v = √(GM/r).
Earth's orbital velocity around the Sun (r ≈ 1 AU): ~29.8 km/s.
ISS orbits at r ≈ R + 400 km with v ≈ 7.7 km/s, T ≈ 92 minutes.
Geostationary orbit: must orbit eastward at Earth's rotational period (~24 hr) in the equatorial plane.
Orbital velocity doesn't depend on the orbiting body's mass — same speed for a satellite or a tennis ball.
Lowest orbit possible is just above atmosphere — below this, drag pulls it down.
Orbital Velocity notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.