Question

What is the minimum energy required to launch a satellite of mass m from the surface of a planet of mass M and radius R in a circular orbit at an altitude of R?

• A. \( \dfrac{5GmM}{6R} \)
• B. \( \dfrac{2GmM}{3R} \)
• C. \( \dfrac{GmM}{2R} \)
• D. (3GmM)/(4R)

Hint:

For a satellite in circular orbit: E = -(GMm)/(2r) Minimum launch energy is calculated using the change in total mechanical energy.

Answer 1

The Correct Option is D

Step 1: Radius of the circular orbit r = R + R = 2R
Step 2: Initial mechanical energy of the satellite at rest on the surface Eᵢ = -(GMm)/(R)
Step 3: Final mechanical energy of the satellite in a circular orbit Ef = -(GMm)/(2r) = -(GMm)/(4R)
Step 4: Minimum energy required to launch the satellite Δ E = Ef - Eᵢ = (-(GMm)/(4R)) - (-(GMm)/(R)) = (3GMm)/(4R)