Let t be the time taken for the two masses to collide and x5M,xM be the distance travelled by the mass 5M and M respectively. The gravitational force acting between two sphere when the distance between them ( 12R−x) where x is a variable, F=
GM×5M
(12R−x)2
Acceleration of mass M,aM=
G×5M
(12R−x)2
Acceleration of mass 5M,a5M=
GM
(12R−x)2
For mass 5M u=0,S=x5M,t=t,a=a5M S=ut+
1
2
at2 ∴x5M=
1
2
a5Mt2.....(i) For mass M u=0,s=xM,t=t,a=aM ∴s=ut+
1
2
at2⇒ xM=
1
2
aMt2....(ii) Dividing (ii) by (iii)
x5M
xM
=
1
2
a5Mt2
1
2
aMt2
=
a5M
aM
=
1
5
∴5x5M=xM.......(iii) From the figure, x5M+xM=12R−2R−R=9R......(iv) From (iii) and (iv)
xM
5
+xM=9R ∴6xM=45R ∴xM=
45
6
R=7.5R
So two sphere collide when the sphere of mass M covered the distance of 7.5R.