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 2R?

• A. (5GMm)/(6R)
• B. (2GMm)/(3R)
• C. (GMm)/(2R)
• D. (GMm)/(2R)

Hint:

For circular orbits: Total Energy = -(GMm)/(2r) Minimum launch energy is always: Ef - Eᵢ where initial energy is gravitational potential energy on the surface.

Answer 1

The Correct Option is A

Step 1: Initial energy of the satellite on the surface of the planet Eᵢ = -(GMm)/(R) 

Step 2: The satellite is placed in a circular orbit at height 2R, so orbital radius r = R + 2R = 3R Total energy of a satellite in circular orbit: Ef = -(GMm)/(2r) = -(GMm)/(6R) 

Step 3: Minimum energy required Δ E = Ef - Eᵢ Δ E = (-(GMm)/(6R)) - (-(GMm)/(R)) Δ E = (5GMm)/(6R)