Three balls A, B and C are placed on a smooth horizontal surface. Given that${\text{m}}_{\text{A}}={\text{m}}_{\text{C}}=4{\text{m}}_{\text{B}}$ . Ball B collides with ball A with an initial velocity V as shown in figure. Find the total number of collisions between the balls. All collisions are elastic. 

Case – 1 :- 1^st collision

Given that${\text{m}}_{\text{A}}=4\text{m,}{\text{m}}_{\text{B}}=\text{m,}{\text{m}}_{\text{C}}=4\text{m}$

${\text{m}}_{\text{1}}=\text{m}$

${\text{m}}_2=4\text{m}$

${\text{v}}_1=\frac{\left({\text{m}}_{\text{1}}-{\text{m}}_{\text{2}}\right){\text{u}}_{\text{1}}}{\left({\text{m}}_{\text{1}}+{\text{m}}_{\text{2}}\right)}+\frac{{\text{2m}}_{\text{2}}{\text{u}}_{\text{2}}}{\left({\text{m}}_{\text{1}}+{\text{m}}_{\text{2}}\right)}$

$=\frac{\left(\text{m}-4\text{m}\right)\text{v}}{5\text{m}}+0$

${\text{v}}_{\text{1}}=\frac{-3}{5}\text{v}$

                        

After collision : -

Case – 2 : -

Let are velocities after 2^nd collision.

after 1^st collision  after second collision

So, there is no further collisions

Only 2 collisions take place