Question

The ratio of energy required to raise a satellite of mass \( m \) to a height \( h \) above the earth's surface to that required to put it into the orbit at same height is

• A. \( \frac{h}{R} \)
• B. \( \frac{4h}{R} \)
• C. \( \frac{3h}{R} \)
• D. \( \frac{2h}{R} \)

Hint:

The energy to raise a satellite is more than the energy required to put it into orbit because it involves overcoming gravitational potential energy at every point.

Answer 1

The Correct Option is D

Step 1: Energy to raise the satellite.
The energy required to raise a satellite to height \( h \) above the earth's surface is the work done against gravitational force. This is given by: \[ W = \frac{G M m}{R^2} \times h \] where \( G \) is the gravitational constant, \( M \) is the mass of the earth, \( m \) is the mass of the satellite, and \( R \) is the radius of the earth.
Step 2: Energy to put the satellite into orbit.
To put the satellite into orbit, the energy required is: \[ E_{\text{orbit}} = \frac{GMm}{2R} \]
Step 3: Ratio of energies.
The ratio of energy required to raise the satellite to the energy required to put it into orbit is: \[ \frac{W}{E_{\text{orbit}}} = \frac{2h}{R} \]
Step 4: Conclusion.
The ratio is \( \frac{2h}{R} \), so the correct answer is (D).