An object of mass m moves with velocity $v$ towards a stationary object of same mass. Impact is an elastic collision.
$v_1$ is the velocity after impact of the mass originally moving
$v_2$ is the velocity after impact of the mass originally stationary
Elastic collision means K.E. is conserved, so: $$\frac{1}{2}mv^2=\frac{1}{2}m(v_1)^2+\frac{1}{2}m(v_2)^2$$ Thus, $$v^2=(v_1)^2+(v_2)^2$$
However, using relative velocities:
$$-v=(v_1)-(v_2)$$
Squaring both sides gives a different value for $v^2$. Instead of $v^2=v_1^2+v_2^2$, $v^2=v_1^2-2v_1v_2+v_2^2$. How come?