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JEE MAIN 2023
11-4-2023
Question
A spaceship of mass $2 \times 10^4 \mathrm{~kg}$ is launched into a circular orbit close to the earth surface. The additional velocity to be imparted to the spaceship in the orbit to overcome the gravitational pull will be (if g = 10 m/s2 and radius of earth = 6400 km)
Select the correct option:
A
$11.2(\sqrt{2}-1) \mathrm{km} / \mathrm{s}$
B
$7.9(\sqrt{2}-1) \mathrm{km} / \mathrm{s}$
C
$8(\sqrt{2}-1) \mathrm{km} / \mathrm{s}$
D
$7.4(\sqrt{2}-1) \mathrm{km} / \mathrm{s}$
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
Sol. $$ \begin{aligned} & \mathrm{v}_{\text {orbit }}=\sqrt{\frac{\mathrm{GM}}{\mathrm{R}}}=\sqrt{\mathrm{gR}} ; \\ & \mathrm{v}_{\text {escape }}=\sqrt{\frac{2 \mathrm{GM}}{\mathrm{R}}}=\sqrt{2 \mathrm{gR}} \\ & \Delta \mathrm{v}=(\sqrt{2}-1) \sqrt{\mathrm{gR}}=8(\sqrt{2}-1) \mathrm{km} / \mathrm{s} \end{aligned} $$
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