Two uniform soild spheres of equal radii `R` but mass `M` and `4M` have a centre to centre separation `6 R`, as shows in Fig. (a) 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 centre of teh second. Obtain an expression for the minimum speed `upsilon` of the projectile so that it reaches the surface of second sphere.

Explanation

The neutral point `N` is at a distance `x` from mass `M`, given by
`x=d/(sqrt((m_(2))/(m_(1)))+1)=(6R)/(sqrt((4M)/M)+1)=2R`
The energy at the surface of `M` is `E_(t)=1/2mv^(2)-(GMm)/R-(4GMm)/(5R)`
At the neutral point `N`, the speed approaches zero. the mechanical energy at `N` is purely potential.
`E_(N)=-(GMm)/(2R)-(4GMm)/(4R)`
From the principle of conservation of mechanical energy
`1/2v^(2)(GM)/2-(4GM)/(5R)=-(GM)/(2R)-(GM)/R`
or `v^(2)=(2GM)/R(4/5-1/2)rArr v=((3GM)/(5R))^(1//2)`