An astronaut in a circular Earth orbit observₑ a body in a lower coplanar circular orbit moving in the same direction. At closest approach the body's speed relative to the astronaut is 5 m s⁻¹, while the astronaut's speed is 5000 m s⁻¹. If the minimum separation is α km, find α/4. Take Earth's mass = 6 × 10²⁴ kg and round α to the nearest integer.
Physics · Gravitation · Satellites, Orbital Motion & Kepler’s Laws
JEE Mains Advanced

Question

An astronaut in a circular Earth orbit observₑ a body in a lower coplanar circular orbit moving in the same direction. At closest approach the body's speed relative to the astronaut is 5 m s⁻¹, while the astronaut's speed is 5000 m s⁻¹. If the minimum separation is α km, find α/4. Take Earth's mass = 6 × 10²⁴ kg and round α to the nearest integer.

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

G1 Physics · Accepted Answer

Step 1: Circular speed v=√(GM/r), so r=GM/v². The lower body's speed is 5005 m s⁻¹ at closest approach because it movₑ 5 m s⁻¹ faster in the same direction. Step 2: rₐ-rᵦ=GM(1/5000²-1/5005²). With GM=G(6×10²⁴)≈4.0×10¹⁴ SI, the separation is about 32 km. Step 3: α≈32, so α/4=8.