Two spherical stars A and B initially have the same radius, densities ρₐ and ρᵦ, and masses related by Mᵦ = 2Mₐ. Star A loses mass while retaining density ρₐ so that its radius becomes half. All lost mass is deposited as a thick shell on B with density ρₐ. If vₐ and vᵦ are the final escape speeds and vᵦ/vₐ = \sqrt{} (10ⁿ/15¹ᐟ³), find n.
Physics · Gravitation · Acceleration Due to Gravity & Escape Velocity
JEE Advanced 2022

Question

Two spherical stars A and B initially have the same radius, densities ρₐ and ρᵦ, and masses related by Mᵦ = 2Mₐ. Star A loses mass while retaining density ρₐ so that its radius becomes half. All lost mass is deposited as a thick shell on B with density ρₐ. If vₐ and vᵦ are the final escape speeds and vᵦ/vₐ = \sqrt{} (10ⁿ/15¹ᐟ³), find n.

Physics · Gravitation · Acceleration Due to Gravity & Escape Velocity

JEE Advanced 2022

G1 Physics · Accepted Answer

Step 1: Initially equal radii and Mᵦ=2Mₐ imply ρᵦ=2ρₐ. After A shrinks to R/2 at the same density, its mass becomes Mₐ/8, so lost mass=7Mₐ/8. Step 2: This mass forms a shell on B at density ρₐ; use volume conservation to find B's new outer radius, then ve=√(2GM/Rsurface) for each final star. Step 3: The algebra givₑ the ratio in the stated form with n=2.30.