Can u derive the formula for escape velocity with proper explanation?

1 Answer
Mar 31, 2018

#=> v = sqrt((2GM)/r)#

Explanation:

Escape velocity is the minimum threshold velocity required by an object to escape from the surface of a body and overcome its gravitational force.

Another way of thinking about it is with energy. Since kinetic energy is dependent on the square of the velocity, an object with enough kinetic energy can overcome the gravitational work being done by an object.

Kinetic energy of an object is defined as:

#=>E_k = 1/2 mv^2#
where #m# is the mass of the object and #v# is the velocity of the object.

Gravitational potential energy is given as:

#=> E_g = (GMm)/r#
where #G# is the gravitational constant, #M# is the mass of the body to escape from, #m# is the mass of the object trying to escape, and #r# is the distance between the objects.

In order to escape, the minimum required kinetic energy must just exceed the gravitational potential energy. So let's set them equal:

#=> E_k = E_g#

#=> 1/2 mv^2 = (GMm)/r#

#=> 1/2 cancel m v^2 = (GMcancelm)/r#

#=>v^2 = (2GM)/r#

Hence:

#=> v = +- sqrt((2GM)/r)#

If the magnitude of this velocity is all that is wanted, then just take the positive result:

#=> v = sqrt((2GM)/r)#