7.2.5

# Escape Velocity & Synchronous Orbits

Test yourself

## Escape Velocity

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

### Equations for escape velocity

• The escape velocity, ve, 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, RP is the radius of the planet and m is the mass of the satellite.

### 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

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

## Synchronous Orbits

A geosynchronous orbit has a period of exactly one day.

### 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

• 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.