Question

A body is projected vertically upwards with a speed of $\sqrt{\frac{GM}{R}}$. The body will attain a height of
$(M$ is mass and $R$ is Radius of earth$)$

• A. $\frac{R}{2}$
• B. $R$
• C. $\frac{5}{4}R$
• D. $\frac{3R}{2}$

Answer 1

The correct option is B $R$

Since, there are no external or non-conservative forces acting on the system which is (body + earth), work done by them would be zero and hence mechanical energy of the system will be conserved.

Let $m$ be the mass of the body.

Therefore, initial mechanical energy is given as

$E_i=U_i+K_i=-\frac{GMm}{R}+\frac{m{v}^2}{2}$

$\Rightarrow E_i=-\frac{GMm}{R}+\frac{1}{2}m(\frac{GM}{R})$

$[\because$ initial velocity is given as $\sqrt{\frac{GM}{R}}]$

$\Rightarrow E_i=-\frac{GMm}{2R}$

And, final mechanical energy will be,

$E_f=U_f+K_f$

$\Rightarrow E_f=-\frac{GMm}{r}+0=-\frac{GMm}{r}$

$[r$ is the maximum distance from the center of the Earth. At the highest point, velocity of the particle will be zero$]$

Conserving mechanical energy at earth surface and at the maximum height attained by the body,

$P{E}_i+K{E}_i=P{E}_f+K{E}_f$

$\Rightarrow -\frac{GMm}{2R}=-\frac{GMm}{r}$

$\Rightarrow r=2R\Rightarrow R+h=2R$

$\therefore h=R$

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

Key Concept - The gravitational potential energy for $2$ point masses is given by $U=-G{m}_1{m}_2/r$ Why this question? Note - A uniform spherical mass can be considered to be a point mass concentrated at its center to calculate gravitational potential energy, if the other mass is outside the sphere.