Escape Velocity & Synchronous Orbits

Physics: AQA A Level, topic 7.2.5

Escape Velocity

Escape velocity is the velocity needed for an object to escape a planet's gravitational pull.

Equations for escape velocity

Equations for escape velocity

The escape velocity, v_e, needed for an object to leave the gravitational influence of a planet can be estimated by equating the expressions:
$KE = \frac12 mv^2$
$GPE = m\Delta V = 0 - (-\frac{GMm}{R_P}) = \frac{GMm}{R_P}$
Where M is the mass of the planet, R_P is the radius of the planet and m is the mass of the satellite.

Calculation for escape velocity

Calculation for escape velocity

When an object is removed from a planet’s gravitational pull, it loses kinetic energy equal to the gravitational potential energy it gains.
The kinetic energy lost is equal to the potential energy gained when the object is moved an infinite distance from the planet and has zero velocity.
$\frac12 m v_e^2=\frac{GMm}{R_P}$
$v_e^2 = \frac{2GM}{R_P}$
$v_e = \sqrt{\frac{2GM}{R_P}}$

Earth's escape velocity

Earth's escape velocity

For the Earth, R_P = 6.4 × 10⁶ m and M = 6.0 × 10²⁴ kg.
This gives v_e of approximately 11 km/s.

Synchronous Orbits

A geosynchronous orbit has a period of exactly one day.

Satellites

Satellites

A satellite in a geosynchronous orbit remains at the same point above the Earth at all times.
These satellites can be used for weather mapping and observation as they can watch the same place for long periods of time.

Calculation

Calculation

By definition, a geosynchronous orbit has a period of one day, this can be used to calculate the radius of the orbit.
$r^3=GmT^2/4{\pi}^2$
Remember to convert the period into seconds and then, take the cubed root to find r.

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