Question

Two stars each of mass $M$ are approaching each other for a head-on collision. When they are at a distance $r$, their speeds are negligible. The radius of both stars is $R$ $(r>>R) .$ What is the speed with which they collide?

• A. $\left(\frac{G M}{R}\right)^{\frac{1}{2}}$
• B. $\left(\frac{G M}{2 R}\right)^{\frac{1}{2}}$
• C. $\left(\frac{G M}{4 R}\right)^{\frac{1}{2}}$
• D. $\left(\frac{2 G M}{R}\right)^{\frac{1}{2}}$

Answer 1

Answer: (B)

$U_{i}+K_{i}=U_{f}+K_{f}$
$\frac{-G M^{2}}{r}+0=\frac{-G M^{2}}{2 R}+2\left(\frac{1}{2} M v^{2}\right)$
$G M^{2}\left[\frac{1}{2 R}-\frac{1}{r}\right]=M v^{2} \Rightarrow v^{2}=\frac{G M(r-2 R)}{2 R r}$
As $(r>>R),(r-2 R)=r \Rightarrow v=\left(\frac{G M}{2 R}\right)^{\frac{1}{2}}$