Escape velocity

Escape velocity (v e ) = (2GM/R)1/2 = (2Rg o )1/2

The escape velocity of a planet or indeed any other gravitational system is the velocity that it would have to be given to escape from the gravitational field of that planet. It could be a provided in a short burst of acceleration to gain enough kinetic energy and would be unpowered after that.If we consider a space probe of mass m at the surface of a planet radius R then :Gravitational potential energy = -GMm/R the energy required to escape from the field is therefore +GMm/R.Kinetic energy to be applied = ½ mvand so ½ (mv) = GMm/R. Therefore :

It is interesting to think about the maximum velocity that might be reached by a meteorite falling onto the Earth's surface. If the meteorite starts from infinity with zero velocity and is accelerated towards the Earth by the Earth's field alone then you should be able to see that the greatest speed that it could have when it reached the Earth is the escape velocity of the Earth.

Example problems

1. Calculate the escape velocity of the Earth given that g o = 9.8ms-2 and that the radius of the Earth is 6.4x106 m



Escape velocity ve = (2Rg o ) = (2x6.4x106x9.8) = 1.12x104 ms-1 = 11.2 kms-1



2. Calculate the radius of a planet with the an escape velocity of 6.2 ms-1 - about the speed of take off of a top class high jumper on Earth. Assume the planet has the same average density as the Earth - 5500 kgm-3.

Escape velocity v e = (2GM/R) therefore R = 2GM/v e 2 =2G(4/3) p R3r/v e 2



and so R = (v e 2/2G(4/3) p r)1/2

Planet radius = (6.22/2x6.67x10-11x1.33x p x5500) = 3.53x103 m = 3.53 km!

Change in g with height

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The escape velocity is also important if we want to find whether a planet can retain its atmosphere. The higher velocity molecules will escape the gravitational pull if their velocity is greater than the escape velocity of the planet.This is minimal until satellite orbit heights are reached. In fact for a height of some 200 km above the Earth's surface the g value has decreased from about 9.8 msto around 7.5 ms