Suppose 2 sphere of mass 1Kg move with velocity 5m/s and 10m/s in same direction. Let us call the ball moving with 5m/s blue and other one red with blue ahead of red from origin.
Initial KE of Blue = 12.5 ; Initial KE of Red = 50
Now, assuming elastic head on collision and by conservation of momentum,
$1×5 + 1×10 = 1×u₁ - 1×u₂$ (the one which moves ahead is u₁ and the one which turns back is u₂)
also by conserving KE,
Blue: 0.5×(25 −u₁²)---(a)
and Blue+Red = 0.5×(25+100) or $u₂² = u₁²$ I could solve this by using the above 2 equation.
Now since I have not explicitly mentioned that u₁ is new velocity of blue and u₂ of red, I could easily replace u₁ and u₂ in (a) and (b)
Which won't change the numerical value of u₁ and u₂ but suppose if the balls were indistinguishable i.e point size of same colour, how would I know which one picked u₁ and which one u₂