I have two particles, one of them is stationary, another one was speed $u$ in the lab's frame. I cna prove with Lorentz transformations that the speed of the zero momentum frame is
$$v=\frac{\gamma_u}{1+\gamma_u}u$$
I can also prove that if I have a particle with speed $u$, then the speed $u'$ of that particle observed from a different frame with speed $v$ (axes aligned with lab frame, moving along $x$-axis - ie standard Lorentz boost frame) is given by
$$u'=\frac{u-v}{1-uv/c^2}$$
Putting this two together, the particle which is stationary in the lab frame, has the same speed in the ZMF as the ZMF itself has in the lab frame (ie $|u'(u=0|=v$).
How could I see this intuitively, without much calculation?
Why the speed of a lab-frame stationary particle in ZMF is the speed of the ZMF in lab's frame?