A small body of mass is projected with a velocity just sufficient to make it reach from the surface of a planet ( or radius `2R` and mass `3M`) to the surface of another planet (or radius `R` and mass `M`). The distance between the centres of the two spherical planets is `6R`. the distance of the body from the centre of bigger planet is 'x' at any moment. During the journey, find the distance 'x' where the speed of the body is (a) maximum (b) minimum. Assume motion of body along the line joining centres of planets.

Answer

`2R, 3R[3-sqrt(3)]`

Explanation

(b) From minimum velocity of projection to reach other planet body should cross the field of other. If this position velocity of projected body is minimum.
`g_(P)=-(G(3m))/(x^(2))+(Gm)/((6R-x^(2)))`
`0=-(3Gm)/(x^(2))+(GM)/((6R-x^(2)))`
`(3)/(x^(2))=(1)/((6R-x^(2))) rArr 6R-x=+-(x)/(sqrt(3))`
`x+(x)/(sqrt(3))=6R rArr ((sqrt(3)+1)x)/(sqrt(3))=6R`
`x = (6sqrt(3)R)/((sqrt(3)+1)) rArr x=(6sqrt(3)(sqrt(3)-1)R)/(2) rArr x=3(3-sqrt(3))R`