A spacecraft is in a stable circular orbit at an altitude equal to 2R 2R\,2R above the surface of a planet of radius RRR. The acceleration due to gravity at the planet's surface is ggg. The spacecraft's engines are fired to give it a sudden boost in speed in its direction of travel, allowing it to escape the planet's gravitational field completely.
What is the minimum speed boost (additional speed) Δv \Delta v\,Δv required for the spacecraft to escape?
(2−1)gR3(\sqrt{2}-1)\sqrt{\frac{gR}{3}}(2−1)3gR
(2−1)gR2(\sqrt{2}-1)\sqrt{\frac{gR}{2}}(2−1)2gR
2gR3\sqrt{\frac{2gR}{3}}32gR
gR3\sqrt{\frac{gR}{3}}3gR
