Question

An artificial satellite of mass m is moving in a circular orbit at a height equal to radius R of earth, suddenly due to the explosion, it breaks into two parts of equal pieces. One part stops just after the explosion. Determine the Increase in mechanical energy of the system due to the explosion is

Answer 1

$\frac{v^2}{2R}=\frac{g}{4}$

V = velocity of satellite

$\frac{g}{4}$ = gravitational accleration at height R from earth surface

$v=\sqrt{\frac{gR}{2}}$

after explosion momentum is conserve and vet u is the velocity after explosion

$mv=\frac{m}{2}\times 0+\frac{m}{2}uu=2v=\sqrt{2gR}$

Increase in mechanical energy = kinetic energy final - kinetic energy initial

$=\frac{1}{2}m{v}^2-\frac{1}{2}\left(\frac{m}{2}\right)u^2=\frac{1}{2}{mv}^2-\frac{m}{4}{4v}^2=\frac{1}{4}{mv}^2=\frac{m}{4}\left(\frac{gR}{2}\right)=\frac{mgR}{8}$