Question

If an angle is projected vertically upwards with speed half the escape speed of earth, then the maximum height attained by it is [R is radius of earth]

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

Answer 1

The correct option is D $\frac{R}{3}$

We know escape velocity of earth $=v_e=\sqrt{2gR}$

$\frac{1}{2}m{v}_e^2-\frac{GMm}{R}=0$ $[KE+PE]=0$

When body is projected from earth's surface

Initial KE$=\frac{1}{2}m{u}^2=mh\frac{R}{4}=\frac{GMm}{4R}$

Initial PE$=\frac{-GMm}{R}$

$\Rightarrow$ Total energy $=\frac{-3GMm}{4R}$

From conservation of energy

$\frac{-GMm}{R+h}=\frac{-3Gm}{4R}$

$3(R+h)=4R$

$\Rightarrow h=\frac{R}{3}$.