Gravitation Part 1

Let $t$ be the time taken for the two masses to collide and $x_{5 M}, x_M$ be the distance travelled by the mass $5 M$ and $M$ respectively. The gravitational force acting between two sphere when the distance between them ( $12 R - x)$ where $x$ is a variable, $F = \;\frac{G M \times 5 M}{\left(12 R - x\right)^2}$ Acceleration of mass $M, a_M = \;\frac{G \times 5 M}{\left(12 R - x\right)^2}$ Acceleration of mass $5 M, a_{5 M} = \;\frac{G M}{\left(12 R - x\right)^2}$ For mass $5 M$ $u=0, S=x_{5 M}, t=t, a=a_{5 M}$$S = u t + \;\frac{1}{2} a t^2$$\therefore x_{5 M} = \;\frac{1}{2} a_{5 M} t^2$.....(i) For mass $M$ $u=0, s=x_M, t=t, a=a_M$$\therefore s = u t + \;\frac{1}{2} a t^2 \Rightarrow$ $x_M = \;\frac{1}{2} a_M t^2$....(ii) Dividing $(i i)$ by $(i i i)$$\;\;\frac{x_{5 M}}{x_M} = \frac{\;\frac{1}{2} a_{5 M} t^2}{\frac{1}{2} a_M t^2}$$\;= \frac{a_{5 M}}{a_M} = \;\frac{1}{5}$$\therefore 5 x_{5 M} = x_M$.......(iii) From the figure,$x_{5 M} + x_M = 12 R - 2 R - R = 9 R$......(iv)From $(i i i)$ and $(i v)$$\;\frac{x_M}{5} + x_M = 9 R$$\therefore 6 x_M = 45 R$$\therefore x_M = \;\frac{45}{6} R = 7.5 R$ So two sphere collide when the sphere of mass $M$ covered the distance of $7.5 R$.