Question

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

• A. \( \frac{G Mm}{6R} \)
• B. \( \frac{G Mm}{2R} \)
• C. \( \frac{G }{6RMm} \)
• D. \( \frac{G Mm}{3R} \)

Hint:

When calculating energy changes in orbital mechanics, use the formula for mechanical energy of the satellite and the difference in potential energy at different radii.

Answer 1

The Correct Option is A

Step 1: Use the formula for mechanical energy of satellite.
The mechanical energy \( M.E \) of the satellite is given by: \[ M.E = -\frac{G Mm}{2a} \] Where \( a \) is the semi-major axis of the orbit.
Step 2: Calculate the work done.
The work required to move the satellite from radius \( 1.5R \) to \( 3R \) is given by: \[ W = \Delta M.E = M_f - M_i \] \[ W = -\frac{G Mm}{2(3R)} + \frac{G Mm}{2(1.5R)} \] Simplifying the expression: \[ W = \frac{G Mm}{R} \left( \frac{1}{6} + \frac{1}{3} \right) \] \[ W = \frac{G Mm}{6R} \] Step 3: Conclusion.
The additional minimum energy required is \( \frac{G Mm}{6R} \).