The Juggler Problem

[essentialsaltes]

This method also addresses your ongoing lying that I did not consider the Earth’s momentum.
Since Vi = Vf = 0, the Earth’s momentum is zero during an elastic collision and therefore does not turn up in the conservation equations.

In the COM frame using the transformation v → v – mv/(M+m) :
Initial momentum of ball = m(v-mv/(M+m)) = mv - m²v/(M+m)
Initial momentum of Earth = -Mmv/(M+m)
...
Final momentum of Earth = Mmv/(M+m)

Glad we cleared up who was lying.

At least now you're... trying... to deal with the Earth. And have discovered that its motion is different before and after, rather than the same.