I am practicing a conservation of momentum question:
Two identical buggies move one after the other due to inertia (without friction) with the same velocity $v_0$. A man of mass m rides the rear buggy. At a certain moment the man jumps into the front buggy with a velocity u relative to his buggy. Knowing that the mass of each buggy is equal to M, find the velocities with which the buggies will move after that.
In the solution for this question mentioned in my book,
Conservation of momentum in rear buggy: $(M+m)v_0 = m[u +v_r] + Mv_r$,
where velocity of man with respect to ground is $u+v_r$ and final velocity of rear buggy is $v_r$.
My question is, why is velocity of man with respect to ground given as $u+v_r$? Shouldn't it be $u + v_0$? Isn't relative velocity here supposed to be taken from before the man starts moving?
Why have they considered the relative velocity of man with respect to the final velocity of buggy? (I don't think their answer is incorrect, though, a few other online sources detail a similar working).