Satellites and Escape Velocity

Escape velocity — the minimum speed an object needs at a planet's surface to escape its gravitational field entirely, with no further propulsion.

Deriving it — the object must be given enough kinetic energy to raise its total energy to zero, since zero total energy corresponds to just reaching infinity with no speed left:

½ m v² = G M m / r

The object's mass cancels, giving

v = √(2 G M / r)

Example: for the Earth, v = √(2 × 6.67 × 10⁻¹¹ × 5.97 × 10²⁴ ÷ 6.37 × 10⁶) = 1.12 × 10⁴ m s⁻¹, about 11.2 km s⁻¹ or 25 000 mph.

The mass cancels — escape velocity is the same for a pebble and a spacecraft. It depends only on the planet's mass and radius.

Why rockets do not reach escape velocity at launch — escape velocity is the speed needed for an unpowered projectile given all its energy at once. A rocket keeps its engines running, so it climbs steadily and never needs that speed. The concept applies to something thrown, not something driven.

Geostationary orbits — a satellite that stays above the same point on the Earth's surface. Three conditions must hold together:

a period of exactly 24 hours, matching the Earth's rotation,

an orbit directly above the equator, and

motion west to east, in the same direction as the Earth's rotation.

The radius follows from the period — substituting T = 86 400 s into T² = (4π²/GM)r³ gives r ≈ 4.2 × 10⁷ m from the Earth's centre, about 36 000 km above the surface. Every geostationary satellite must be at that one altitude, which is why the equatorial belt is a congested and regulated resource.

Uses — geostationary satellites suit communications and television, since a ground aerial can be aimed once and left. Their drawback is the distance: signals take about a quarter of a second for the round trip, and the poles are poorly covered.

Low polar orbits — much closer, with periods around 90 minutes, passing over different strips of the surface as the Earth turns beneath. Used for imaging, weather and surveillance, where coverage of the whole surface matters more than a fixed position.