A perfectly elastic ball ${\text{P}}_1$  of mass m moving with velocity v collides elastically with three exactly similar balls${\text{P}}_{\text{2}}{\text{,P}}_{\text{3}}{\text{,P}}_{\text{4}}$ lying on a smooth table as shown. Velocities of the four balls after the collision are

${\text{u}}_1=\text{v,}{\text{u}}_{\text{2}}={\text{u}}_{\text{3}}={\text{u}}_{\text{4}}=0$

In first case:

${\text{v}}_{\text{1}}=\frac{\left({\text{m}}_1-{\text{m}}_2\right){\text{u}}_1}{\left({\text{m}}_{\text{1}}+{\text{m}}_2\right)}+\frac{{\text{2m}}_{\text{2}}{\text{u}}_{\text{2}}}{\left({\text{m}}_{\text{1}}+{\text{m}}_{\text{2}}\right)}$

$=\frac{\left(0\right)\text{v}}{2\text{m}}+\frac{\text{2m}\left(\text{0}\right)}{\text{2m}}$

${\text{v}}_{\text{1}}=0$

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

$=0+\frac{\text{2mv}}{\text{2m}}$

${\text{v}}_{\text{2}}=\text{v}$

So, If a body of mass ‘m’ collides elastically with similar body, they exchange their velocities.

${\text{I}}^{\text{st}}\text{case}\to {\text{P}}_{\text{1}}\text{hits}{\text{P}}_{\text{2}}\text{with}\text{'v'}\text{\&}{\text{v}}_{\text{1}}\Rightarrow 0,{\text{v}}_{\text{2}}\Rightarrow \text{v}$

${\text{II}}^{\text{nd}}\text{case}\to {\text{P}}_{\text{2}}\text{hits}{\text{P}}_{\text{3}}\text{with}\text{'v'}\text{\&}{\text{v}}_{\text{2}}\Rightarrow 0,{\text{v}}_{\text{3}}\Rightarrow \text{v}$

${\text{III}}^{\text{rd}}\text{case}\to {\text{P}}_{\text{3}}\text{hits}{\text{P}}_{\text{4}}\text{with}\text{'v'}\text{\&}{\text{v}}_{\text{3}}\Rightarrow 0,{\text{v}}_{\text{4}}\Rightarrow \text{v}$

So velocities of four balls after collision 0, 0, 0, v