Question

A triple system consists of three stars of mass m each at corners of an equilateral triangle of side R. Find the escape velocity of one of the stars

• A. $v=2\sqrt{\left(Gm/R\right)}$
• B. $v=\sqrt{\left(Gm/R\right)}$
• C. $v=\sqrt{\left(Gm/2R\right)}$
• D. $v=\sqrt{\left(Gm/4R\right)}$

Answer 1

The correct option is A $v=2\sqrt{\left(Gm/R\right)}$

Let escape velocity is $v$.

Gravitational potential energy of the star due to other star $PE=-\frac{G{m}^2}{R}-\frac{G{m}^2}{R}=-\frac{2G{m}^2}{R}$,

Kinetic energy of the particle $KE=\frac{1}{2}m{v}^2$

$\therefore$ Total Energy $=PE+KE=-\frac{2G{m}^2}{R}+\frac{1}{2}m{v}^2$

The total energy of the particle should be zero, if it had to escape from the surface of the earth. Hence, Total Energy $=-\frac{2G{m}^2}{R}+\frac{1}{2}m{v}^2=0$ $\Rightarrow V_e=\sqrt{\frac{4Gm}{R}}=2\sqrt{\left(Gm/R\right)}$

The correct option is (a)