Why can't the collision be elastic if after the collision the two objects move together with the same velocity? I don't quite understand why two bodies can't 'stick' together in elastic collision. I have found several sources that say that if two objects collide and end up moving at the same velocity their collision is perfectly inelastic. I don't get why :( 
 A: Let's assume the bodies do stick together and move with final velocity $v$, and also assume that kinetic energy is conserved. Let's also just work with a collision in one dimension.
Then we have our two conservation equations
$$m_1v_1+m_2v_2=(m_1+m_2)v$$
$$\frac12m_1v_1^2+\frac12m_2v_2^2=\frac12(m_1+m_2)v^2$$
Rearranging we get:
$$m_1v_1-m_1v=m_2v-m_2v_2$$
$$m_1v_1^2-m_1v^2=m_2v^2-m_2v_2^2$$
Dividing the second by the first:
$$\frac{m_1v_1^2-m_1v^2}{m_1v_1-m_1v}=\frac{m_2v^2-m_2v_2^2}{m_2v-m_2v_2}$$
This way the masses cancel, and we can use the fact that $(a^2-b^2)=(a+b)(a-b)$ to arrive at
$$v_1+v=v+v_2$$
Or
$$v_1=v_2$$
But this is a contradiction! If the objects started off with the same velocity and not touching, then there never could have been a collision. Or, they were already touching to begin with. Therefore this is a proof by contradiction showing that energy cannot be conserved when the objects sick together during a collision. The contrapositive is then also true: if kinetic energy is conserved then the objects cannot stick together. In other words, both of our initial assumptions cannot both be true at the same time 
A: By definition, elastic collision is one in which total mechanical energy is conserved. That means kinetic energy of the system before and after collision will be same.
Other way to look at it is using coefficient of restitution. If coefficient of restitution is 0, there is zero recoiling (bouncing back) and the collision is called perfectly inelastic. If the coefficient of restitution is anything greater than 0, it will push the balls (or objects) away from each other upon collision and both the objects can not have same velocity.
You might like to see this - https://www.google.co.in/url?sa=t&source=web&rct=j&url=https://m.youtube.com/watch%3Fv%3D8ko3qy9vgLQ&ved=2ahUKEwjNnvn059feAhUKUI8KHb59D9AQt9IBMA56BAgHEDY&usg=AOvVaw3TiCtJvtPRhMntkq0Cmryf
