We can solve the problem by using the law of conservation of energy.
When the rocket starts its motion from the Earth surface, its mechanical energy is sum of kinetic energy and gravitational potential energy:
where
m is the rocket's mass
is the rocket initial speed
is the gravitational constant
is the Earth's mass
is the distance of the rocket from the Earth's center (so, it corresponds to the Earth's radius)
The mechanical energy of the rocket when it is very far from the Earth is just kinetic energy (because the gravitational potential at infinite distance from Earth is taken to be zero):
where
is the final speed of the rocket.
By equalizing the initial energy and the final energy, we can find the final velocity:
When the rocket starts its motion from the Earth surface, its mechanical energy is sum of kinetic energy and gravitational potential energy:
where
m is the rocket's mass
The mechanical energy of the rocket when it is very far from the Earth is just kinetic energy (because the gravitational potential at infinite distance from Earth is taken to be zero):
where
By equalizing the initial energy and the final energy, we can find the final velocity: