Q: A particle is fired vertically upwards from the surface of earth reaches a height 6400km. Find the initial velocity of the particle.

Sol: TE. on the surface of the earth = TE. at the highest point

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

given, h = R = 6400 km

$\large v^2 = g h $

$\large v = \sqrt{g h} $

$\large v = \sqrt{10 \times 6400 \times 10^3} $

= 8 km/s