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

To solve this problem, we need to calculate the change in energy required to transfer a satellite from one orbit to another around a planet. The energies involved in both the initial and final orbits will help us find the answer.

Step 1: Understand the energy formula.

The total energy \( E \) of a satellite in an orbit of radius \( r \) is given by:

\(E = -\frac{G M m}{2r}\)

Here, \( G \) is the gravitational constant, \( M \) is the mass of the planet, \( m \) is the mass of the satellite, and \( r \) is the radius of the orbit.

Step 2: Calculate the initial energy.

The initial orbit radius is \( 1.5R \). Thus, the initial energy \( E_1 \) is:

\(E_1 = -\frac{G M m}{2 \times 1.5R} = -\frac{G M m}{3R}\)

Step 3: Calculate the final energy.

The final orbit radius is \( 3R \). Thus, the final energy \( E_2 \) is:

\(E_2 = -\frac{G M m}{2 \times 3R} = -\frac{G M m}{6R}\)

Step 4: Calculate the change in energy.

The change in energy \( \Delta E \), which is the additional energy required to move the satellite to the new orbit, is given by:

\(\Delta E = E_2 - E_1\)

Substituting the energies from Step 2 and Step 3, we have:

\(\Delta E = -\frac{G M m}{6R} - \left(-\frac{G M m}{3R}\right)\)

Simplifying the expression, we get:

\(\Delta E = -\frac{G M m}{6R} + \frac{G M m}{3R} = \frac{G M m}{6R}\)

The additional minimum energy required to transfer the satellite from orbit radius \( 1.5R \) to \( 3R \) is:

\(\frac{G M m}{6R}\)

Thus, the correct option is:

\(\frac{G M m}{6R}\)