Gravitation Part 1

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.