A particle of mass ' $\mathrm{m}$ ' is kept at rest a height $3 \mathrm{R}$ from the surface of earth, where…

A particle of mass ' $\mathrm{m}$ ' is kept at rest a height $3 \mathrm{R}$ from the surface of earth, where ' $R$ ' is radius of earth and ' $M$ ' is the mass of earth. The minimum speed with which it should be projected upward, so that it does not return back is ( $\mathrm{g}=$ acceleration due to gravity on the earth's surface)
  1. $\left[\frac{\mathrm{GM}}{2 \mathrm{R}}\right]^{1 / 2}$
  2. $\left[\frac{\mathrm{gR}}{4}\right]^{1 / 2}$
  3. $\left[\frac{2 g}{R}\right]^{1 / 2}$
  4. $\left[\frac{\mathrm{GM}}{\mathrm{R}}\right]^{1 / 2}$

Solution

$\mathrm{K}_{\mathrm{i}}+\mathrm{U}_{\mathrm{i}}=\mathrm{K}_{\mathrm{f}}+\mathrm{U}_{\mathrm{f}}$ $\frac{1}{2} \mathrm{mv}^2-\frac{\mathrm{GMm}}{4 \mathrm{R}}=0$ $\therefore \mathrm{v}=\left[\frac{\mathrm{GM}}{2 \mathrm{R}}\right]^{1 / 2}$

Asked in: MHT CET 2021 (21 Sep Shift 1)

Practice more Gravitation questions on Aicharya