Escape Velocity

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All bodies when projected vertically upwards fall back to earth due to the earth's gravitational pull. The maximum height attained by a body depends upon the initial velocity of the body. This does not mean that no object can escape earth's gravitational pull. We know that the spacecrafts overcome the earth's gravitational pull and escape into space. That means if a body has to escape into space, it must be propelled with a very high velocity so that it acquires sufficient kinetic energy to overcome the earth's gravitational force. This velocity is referred to as escape velocity (v_esc).

 

The minimum velocity that a moving body must attain in order for it to escape the gravitational pull of the earth (planet) is called escape velocity.

Expression for escape velocity

Consider an object of mass 'm' placed on the surface of the earth. According to Newton's law of gravitation, the gravitational pull of the earth on that object is given by

Where F is the force of attraction between the earth and the object,

M_e is the mass of the earth and R_e is the radius of the earth.

Then, work required to overcome the gravitational pull = F x R_e

If v is the velocity of the projectile (object), then its kinetic energy = 1/2 mv².

For the projectile to escape from earth's gravitational pull the kinetic energy of the projectile has to be greater than the work required to overcome the gravitational pull.

Let us now get an expression for v_esc in terms of acceleration due to gravity.

(Multiply both numerator and denominator by R_e)

.

Substituting for g_e and R_e in equation (2) we get,

Hence, a projectile with a velocity greater than 11.2km/s will escape from the surface of the earth. From the above expression it is clear that escape velocity is independent of the mass of the projectile but depends on the mass and radius of the earth (planet).

Escape velocity of moon

Hence, escape velocity of moon is approximately five times less than that of earth.