Q: A body is projected vertically upwards from the surface of the earth with a velocity equal to half of escape velocity of the earth. If R is radius of the earth, maximum height attained by the body from the surface of the earth is

Sol: From law of conservation of energy,

T.E_{surface} = T.E_{Max Height}

$\large -\frac{G M m}{R} + \frac{1}{2}m (\frac{v_e}{2})^2 = -\frac{G M m}{R+h} + \frac{1}{2}m (0)^2 $

$\large -\frac{G M m}{R} + \frac{1}{2}m (\frac{2GM}{4R}) = -\frac{G M m}{R+h} $

On simplifying, we get h = R/3.