Question

A particle of mass $m$ is kept at rest at a height $3R$ from the surface of earth, where $R$ is radius of earth and $M$ is mass of earth. The minimum speed with which it should be projected, so that it does not return back, is : (g is acceleration due to gravity on the surface of earth)

• A. $\left(\frac{GM}{2R}\right){^{\frac{1}{2}}}$
• B. $\bigg( \frac {gR}{4}\bigg){^{\frac{1}{2}}}$
• C. $\bigg( \frac {2g}{R}\bigg){^{\frac{1}{2}}}$
• D. $\bigg( \frac {GM}{R}\bigg){^{\frac{1}{2}}}$

Answer 1

The Correct Option is A

1. To find the minimum speed required for a particle to not return back to Earth, it must be projected with a velocity that allows it to escape Earth's gravitational pull. This speed is known as the escape velocity.
2. The formula for escape velocity from a planet's surface is given by: v_e = \sqrt{\frac{2GM}{R}}, where G is the gravitational constant, M is the mass of the planet, and R is the distance from the center of the planet.
3. In this scenario, the particle is at a height 3R from the Earth's surface, which means it is at a distance (R + 3R) = 4R from the center of the Earth.
4. The escape velocity from a distance 4R is: v_{e, 4R} = \sqrt{\frac{2GM}{4R}} = \sqrt{\frac{GM}{2R}}.
5. Comparing this result with the given options, the minimum speed with which the particle should be projected so that it does not return back to Earth is: v_{\text{min}} = \sqrt{\frac{GM}{2R}}, which matches the correct answer.
6. Thus, the correct answer is the first option: \left(\frac{GM}{2R}\right)^{\frac{1}{2}}.