Question

With what velocity should a particle be projected upwards so that it reaches a height equal to radius of earth$\left(R\right)$?
(Assume $G$ = gravitational constant and $M$ = mass of earth )

• A. ${\left(\frac{GM}{R}\right)}^{1/2}$
• B. ${\left(\frac{8GM}{R}\right)}^{1/2}$
• C. ${\left(\frac{2GM}{R}\right)}^{1/2}$
• D. ${\left(\frac{4GM}{R}\right)}^{1/2}$

Answer 1

The correct option is A ${\left(\frac{GM}{R}\right)}^{1/2}$

Let the mass of the object be $m$ and the speed of projection $v$.


Since external and non-conservative force are absent, so mechanical energy of the system will be conserved.

$\therefore$ mechanical energy at earth surface = mechanical energy at highest point

$\Rightarrow U_i+K{E}_i=U_f+K{E}_f$

$\Rightarrow \frac{-GMm}{R}+\frac{1}{2}m{v}^2=\frac{-GMm}{(R+R)}+0$

$\because$ at the highest point speed will be zero hence $K{E}_f=0$ and given, $h=R$

$\Rightarrow \frac{-GMm}{R}+\frac{1}{2}m{v}^2=\frac{-GMm}{2R}$

$\Rightarrow \frac{GMm}{2R}=\frac{1}{2}m{v}^2$

$\Rightarrow v=\sqrt{\frac{GM}{R}}$

Hence, option $\left(a\right)$ is correct.