A satellite of mass m moving in a circular Earth orbit of radius r. It loses mechanical energy due to small air resistance at a constant rate C J s⁻¹. If the Earth's radius is Rₑ and the time to spiral down to the Earth is t = [GMm/(nC)](1/Rₑ - 1/r), find n.
Physics · Gravitation · Satellites, Orbital Motion & Kepler’s Laws
JEE Mains Advanced

Question

A satellite of mass m moving in a circular Earth orbit of radius r. It loses mechanical energy due to small air resistance at a constant rate C J s⁻¹. If the Earth's radius is Rₑ and the time to spiral down to the Earth is t = [GMm/(nC)](1/Rₑ - 1/r), find n.

Physics · Gravitation · Satellites, Orbital Motion & Kepler’s Laws

G1 Physics · Accepted Answer

Step 1: Circular-orbit total energy is E(r)=-GMm/(2r). Step 2: Energy lost in going from radius r to Rₑ is |ΔE|=GMm/2(1/Rₑ-1/r). At constant loss rate C, t=|ΔE|/C. Step 3: t=[GMm/(2C)](1/Rₑ-1/r), so n=2.