Two uniform solid spheres of equal radii $R$, but mass $M$ and $4M$ have a centre-to-centre separati

Question

Two uniform solid spheres of equal radii $R$, but mass $M$ and $4M$ have a centre-to-centre separation $6R$. A projectile of mass $m$ is projected from the surface of the sphere of mass $M$ directly towards the centre of the second sphere. What is the minimum speed of the projectile so that it reaches the surface of the second sphere?

Professor PC Thomas Classes · Answer

The projectile must just reach the neutral point where the gravitational pulls cancel. The neutral point is at a distance $2R$ from the centre of mass $M$. Using conservation of energy: $\frac{1}{2}mv^2 - \frac{GMm}{R} - \frac{4GMm}{5R} = -\frac{GMm}{2R} - \frac{4GMm}{4R}$. Solving gives $v = \sqrt{\frac{3GM}{5R}}$.