Question

The total mechanical energy of an object of mass $m$ projected from surface of earth with escape speed is

• A. $$-\frac{GMm}{3R}$$
  
• B. $-\frac{GMm}{2R}$
  
• C. Zero
• D. Infinite

Answer 1

The correct option is C Zero

Given,
Mass of the object$=m$

If $M$ is the mass and $R$ is the radius of the earth, then the escape velocity from its surface is given by,$$v_e=\sqrt{\frac{2GM}{R}}$$$\text{Total mechanical energy (M.E) = K.E + P.E}$

$\Rightarrow \text{M.E =}\frac{1}{2}m{v}_e^2-\frac{GMm}{R}$

Substituting the values of $v_e$,

$\Rightarrow \text{M.E=}\frac{1}{2}m\times \frac{2GM}{R}-\frac{GMm}{R}$

$\Rightarrow \text{M.E =}0$

Hence, option (a) is the correct answer.

Key Concept: The escape velocity for a planet is given by, $$v_e=\sqrt{\frac{2GM}{R}}$$Why this question: To note that the potential energy when the masses are far apart is assumed to be zero.