A satellite is revolving around a planet in orbit radius of 1.5R. Additional minimum energy required to transfer the satellite to new orbit radius of 3R is (m and M are mass of satellite & planet)

Energy required = X (GMm / R)

Find X.

The correct answer is 6.

Step-by-Step Derivation:

The total energy $E$ of a satellite of mass $m$ revolving in a circular orbit of radius $r$ around a planet of mass $M$ is given by the formula:
$E=-\frac{GMm}{2r}$

1. Initial Total Energy ($E_1$):
Initially, the satellite is revolving in an orbit of radius $r_1=1.5R$.
Substituting $r_1$ into the energy formula:
$E_1=-\frac{GMm}{2\left(1.5R\right)}=-\frac{GMm}{3R}$

2. Final Total Energy ($E_2$):
The satellite is transferred to a new orbit of radius $r_2=3R$.
Substituting $r_2$ into the energy formula:
$E_2=-\frac{GMm}{2\left(3R\right)}=-\frac{GMm}{6R}$

3. Additional Energy Required ($\Delta E$):
The minimum energy required to transfer the satellite is the difference between the final and initial total energies:
$\Delta E=E_2-E_1$

Substituting the values:
$\Delta E=-\frac{GMm}{6R}-\left(-\frac{GMm}{3R}\right)$

Simplifying the expression:
$\Delta E=\frac{GMm}{3R}-\frac{GMm}{6R}$

Taking a common denominator:
$\Delta E=\frac{2GMm-GMm}{6R}=\frac{GMm}{6R}$

4. Comparing with the given form:
Assuming the energy required is represented as:
$\text{Energy required}=\frac{GMm}{XR}$

Comparing the derived expression with this form yields:
$X=6$