Maximum speed harmonic motion Consider the following set up:

Masses 1 and 2 are connected by a spring and are at rest. Mass 3 comes with velocity V and strikes ball 1.
I have seen that in this instance, the maximum speed of ball 2 would be twice the center of mass speed. Why is that? If someone can provide an intuitive (or formal) explanation for it I would appreciate it.
*I realize the center of mass speed is constant, and that the motion after the strike is comprised of the center of mass moving and an "internal" harmonic motion.
 A: (Last time I answered a homework question on Physics SE, it was deleted for giving a complete answer to homework-like questions. I will try to take caution in that matter this time.)
Before beginning my answer, I'd like to clarify that while solving this question I did not visualise the setup as "center of mass moving and an "internal" harmonic motion about it". I just focussed on the transfer of momentum from one body to another. So in my solution center of mass does not have much role to play, except at the end when you need to compare the maximum velocity of the mass 2 with the velocity of the center of mass. 
Moving on with the answer:
The first step in getting the solution is to realise how the momentum is being transferred from mass 3 to mass 2. That is, initially the momentum and energy imparted to the system (of mass 1, spring and mass 2) is entirely possessed by mass 1. As mass 1 moves forward, the spring is compressed and mass 2 experiences spring force in forward direction. Mass 1 will keep getting that forward push as long as the spring is in compressed state. 
Then we need to know the situation in which the mass 2 will have the maximum velocity. It would be the point where its acceleration becomes zero, i.e. spring force should be zero, which further implies that the spring, at that moment, would be in realxed state.
Now that we are aware of the situation of the bodies and the spring at that moment, we can get the value of the variables by simply applying the equations of conservation of momentum and conservation of energy. 
MORE HINTS:
Caution: I have written an almost direct solution below, see it only after you have made an actual attempt to solve the question on your own.

 Assume the initial momentum transferred to the system to be $p$. Hence the velocity of center of mass will be $\frac{p}{m_1+m_2}$.

 As this momentum is initially being carried by $m_1$ only, the energy imparted to the system would be $p^2/2m_1$. So we know the total momentum and energy of the system which will be conserved.

 Let the velocity of the two masses be $v_1$ and $v_2$ respectively at the moment when $m_2$ reaches its maximum velocity. Use the two equations (of conservation of momentum and energy) to get $v_2$ and then compare it with the velocity of the centre of mass: $$m_1v_1 + m_2v_2 = p$$ $${\frac 12}m_1v_1^2 + {\frac 12}m_2v_2^2 = \frac{p^2}{2m_1}$$

