Question

An object projected vertically up from the earth's surface with velocity $\sqrt{gR}$, where $R$ is the radius of the earth and $g$ is the acceleration due to gravity on the surface of earth. Find the maximum height reached by the object.

• A. $R$
• B. $2R$
• C. $R/2$
• D. $R/4$

Answer 1

The correct option is A $R$

Let the mass of the object be $m$ and the maximum height reached be $h$.

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

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

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

$\because$ at the highest point speed will be zero hence $K{E}_f=0$

$\Rightarrow \frac{-GMm}{R}+\frac{1}{2}mgR=\frac{-GMm}{R+h}$

Since, $g=\frac{GM}{R^2}$

$\Rightarrow \frac{-GMm}{R}+\frac{1}{2}m\frac{GM}{R}=\frac{-GMm}{R+h}$

$\Rightarrow \frac{1}{2R}=\frac{1}{R+h}$

$\Rightarrow h=R$

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

Why this question: To familiarize students with the application of conservation of mechanical energy, which is a core concept of this chapter.