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Gravitation

JEE Main level, single correct, medium

124 of 223

Question 124 of 223

If earth has a mass nine times and radius twice to the of a planet $P$ . Then $\dfrac{{v}_{e}}{3}\sqrt{x}m{s}^{-1}$ will be the minimum velocity required by a rocket to pull out of gravitational force of $P$ , where ve is escape velocity on earth. The value of $x$ is

Show answer and solution

Answer: (A) $2$

The formula to calculate the escape velocity

\left({v}_{e}\right)

for the Earth is given by

{v}_{e}=\sqrt{\dfrac{2G{M}_{e}}{{R}_{e}}}.......................(1)

where,

G

is the Universal Gravitational constant,

{M}_{e}

is the mass and

{R}_{e}

is the radius of Earth.
The formula for the escape velocity

\left({v}_{esc}\right)

of any planet is given by

{v}_{esc}=\sqrt{\dfrac{2GM}{R}}........................(2)

where,

M

is the mass and

R

is the radius of the planet.
Given that the mass of the planet is

9

times the mass of the Earth and the radius of the planet is

2

times the radius of the Earth.
Substitute

\dfrac{{M}_{e}}{9}

for

M

and

\dfrac{{R}_{e}}{2}

for

R

into equation (2) and simplify to obtain the escape velocity for the planet in terms of that in Earth.

\begin{matrix}{v}_{esc} & = & \sqrt{\dfrac{2G\dfrac{{M}_{e}}{9}}{\dfrac{{R}_{e}}{2}}} \\ & = & \sqrt{\dfrac{2}{9}\dfrac{2G{M}_{e}}{{R}_{e}}} \\ & = & \dfrac{\sqrt{2}}{3}{v}_{e}.....................(3)\end{matrix}

Comparing equation (3) with the given expression, it can be written that

x=2

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More Physics practice

JEE Main Physics practice question on Gravitation (Gravitational potential and Potential Energy), with the answer and a step-by-step solution. Scan a question from your own book or paper to get a whiteboard video.

If earth has a mass nine times and radius twice to the of a planet PPP. Then ve3xms−1\dfrac{{v}_{e}}{3}\sqrt{x}m{s}^{-1}3ve​​x​ms−1 will be the minimum velocity required by a rocket to pull out of gravitational force of PPP, where ve is escape velocity on earth. The value of xxx is

JEE Main question paper PDF

More Physics practice

If earth has a mass nine times and radius twice to the of a planet $P$ . Then $\dfrac{{v}_{e}}{3}\sqrt{x}m{s}^{-1}$ will be the minimum velocity required by a rocket to pull out of gravitational force of $P$ , where ve is escape velocity on earth. The value of $x$ is