[Physics] Why is conservation of momentum not valid here

Question

To explain my confusion, I would provide the following system:

The two masses $m$ and $M$, with $M\gg m$, are moving towards each other(as directed by the arrows) with a common constant speed $V_0$. There is no friction between any two surfaces and all collisions are perfectly elastic.

I take all velocities +ve towards right and I call the velocity of $m$ after collision $V$. As $M\gg m$, there will be negligible change in the velocity of $M$ after collision. Also, as the collision is elastic, the velocity with which the two masses approach each other must be equal to the velocity with which they get separated.

Therefore,
\begin{align}
-V_0-(V_0)&=-V_0-(V)
\\V&=-3V_0
\end{align}

Now, this seems quite true to me.

But, when we apply conservation of momentum,
\begin{align}
mV_0-MV_0&=mV-MV_0
\\V&=V_0
\end{align}

So why is conservation of momentum not valid here?

Answer 1

As $M\gg m$, there will be negligible change in the velocity of $M$ after collision.

Yes, the change in the velocity of $M$ will be negligible, but what is conserved is not the velocity but the momentum and, since $M\gg m$, even a small change in the velocity of $M$ will translate in a relatively big change in its momentum.