A particle is projected from point A, located at distance 4R from the Earth's centre, with speed v₁ making 30° with the radial line. It just grazes the Earth's surface. Considering gravity only, express v₁ as (1000X/√2) m s⁻¹ and find X. Use GM/R = 6.4 × 10⁷ m² s⁻².

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G1 Physics · Accepted Answer

Step 1: At A, r₁=4R and transverse speed is v₁sin30°=v₁/2, so angular momentum per unit mass is h=4R(v₁/2)=2Rv₁. At grazing, r₂=R and radial speed is zero, so v₂=h/R=2v₁. Step 2: Energy conservation givₑ v₁²/2-GM/(4R)=(2v₁)²/2-GM/R. Step 3: Solving givₑ v₁²=GM/(2R), so v₁=√(6.4×10⁷/2)=4000√2=8000/√2 m s⁻¹. Thus X=8.