Question

Two uniform solid spheres of equal radii $R$, but mass $M$ and $4\, M$ have a centre to centre separation $6 \,R$, 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 $\sqrt{\frac{x G M}{y R}}$, then find $(x+y)$.

Answer 1

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{G M m}{r^{2}}=\frac{G(4 M) m}{(6 R-r)^{2}} $
$\Rightarrow r=2 R$
Now by conservation of mechanical energy,
$-\frac{G M m}{R}-\frac{G(4 M) m}{5 R}+\frac{1}{2} m v^{2}$
$=-\frac{G M m}{2 R}-\frac{G(4 M) m}{4 R}+0 $
$\Rightarrow v=\sqrt{\frac{3 G M}{5 R}}$