Question

A satellite of mass m, initially at rest on the earth, is launched into a circular orbit at a height equal to the radius of the earth. The minimum energy required is

Answer 1

U(final) = -GMm/4R

U(initial) = -GMm/R

Where R is radius of earth.

Energy required is U(f)-U(i) = 3GMm/4R = 3mgR/4.

Answer 2

Energy required in minimum from Earth to launch is $\bold{\frac{3 m g R}{4}}$

Solution:

Minimum required energy for satellite to be launched under circular orbit with height being equal to radius of earth, is calculated as follows,

As we know that the energy required to launch is to overcome the energy sustained in the phase of rest, thus the potential energy must be difference of final and initial potential energy.

Initial Potential Energy= $-\frac{G M m}{R}$

Final Potential Energy= $-\frac{G M m}{4 R}$

Minimum Required Energy=Final Potential Energy-Initial Potential Energy

Minimum Required Energy= $-\frac{G M m}{4 R}-(-\frac{G M m}{R})$

Minimum Required Energy= $\bold{\frac{3 G M m}{4 R}}$