question_answer

Three stars each of mass M and radius R are initially at rest and the distance between centres of any two stars is d and they form an equilateral triangle. They start moving towards the centroid due to mutual force of attraction. What are the velocities of the stars just before the collision? Radius of each star is R.

A) \[\sqrt{2GM\left( \frac{d-2R}{5dR} \right)}\]

B) \[\sqrt{\frac{GM(d-2R)}{dR}}\]

C) \[\sqrt{\frac{GM(d-R)}{Rd}}\]

D) \[\sqrt{\frac{GM(d+2R)}{5dR}}\]

Correct Answer: B

Solution :

[b] \[{{E}_{i}}=-\frac{3GMM}{d}\] \[{{E}_{f}}=-3\frac{GMM}{2R}+3\left( \frac{1}{2}M{{v}^{2}} \right)\]                         Initial energy = Final energy \[GM\left[ \frac{1}{2R}-\frac{1}{d} \right]=\frac{1}{2}{{v}^{2}}\] \[v=\sqrt{2GM\left[ \frac{d-2R}{2dR} \right]}\]