Why use momentum instead of kinetic energy for elastic collisions

[tambourine]

if a block in motion hits one at rest, Ek1i = Ek1f + Ek2f

could i also use V1i + V1f = V2i + V2f?

 

[cryptoguy]

No, you can't say that. If you have $$\frac{1}{2}m_1 v_1^2 = \frac{1}{2}m_1 v_{f1}^2 + \frac{1}{2}m_2 v_{f2}^2$$ the most you can do is cancel out the 1/2s because there's no way you can cancel out the masses (or the squares).
You could say $$m_1(v_1^2-v_{f1}^2) = m_2v_{f2}^2$$