Question

Two uniform solid spheres of equal radii $R$, but mass $M$ and $4M$ have a centre to centre separation $6R$, as shown in the figure. The two spheres are held fixed. A projectile of mass $m$ is projected from the surface of the sphere of mass $M$ directly towards the center of the second sphere. If the minimum speed $v$ of the projectile so that it reaches the surface of the second sphere is $\sqrt{\frac{xGM}{yR}}$, then find $(x+y)$.

Answer 1

Answer: 8

The projectile will automatically reach at the surface of the second sphere if we project with a speed which would enable it to reach the neutral point $N$.
Let $r$ be the distance of neutral point $N$ from the sphere of mass $m$.
For neutral point, $\frac{GMm}{r^2}=\frac{G(4M)m}{(6R-r{)}^2}$
$\Rightarrow r=2R$
Now by conservation of mechanical energy,
$-\frac{GMm}{R}-\frac{G(4M)m}{5R}+\frac{1}{2}m{v}^2$
$=-\frac{GMm}{2R}-\frac{G(4M)m}{4R}+0$
$\Rightarrow v=\sqrt{\frac{3GM}{5R}}$