# Relative motion - Cylinders rolling between two identical planks

Today i ran onto this simple problem, which seemed to be interesting to me. Given the illustration bellow, the problem is states as:

Two identical cylinders roll in between two identical planks. If the velocity of each cylinder is $\vec{v}$ and the velocity of the bottom plank is $\vec{u}$ ($|\vec{v}| > |\vec{u}|$), find the velocity the man standing on the top plank needs to obtain with respect to it[the top plank], for which he covers distance $s$ in $t$ seconds in the stationary reference frame ( observer's frame ). Will the velocity vector point right or left? ( $\vec{v}$ and $\vec{u}$ are both given with respect to the stationary reference frame ). All friction is to be neglected and the cylinders are assumed to be rotating without slipping. Let $\vec{v_m}$ be the sought velocity.

Interestingly, a naive approach would be to immediately write $$\vec{v_m} = \frac{\vec{s}}{t} - \vec{v}$$ since one might think that the velocity of the upper plank with respect to the stationary observer is$\vec{v}$.

However, after inspecting the problem more thoroughly, I've come to the following conclusion:

• The velocity of the upper plank comes solely from the rotation of the cylinders.
• Angular velocity of each of the cylinders is$\$ $\omega = \frac{v-u}{R}$

From here it seems that the velocity of the upper plank with respect to the stationary reference frame might be $$\vec{u}' =\vec{v} - \vec{u}$$

And the sought velocity is: $$\vec{v_m} = \frac{\vec{s}}{t} - \vec{u'}$$

Which seems to be ok intuitvely ( the greater the magnitude of $\vec{v}$, the smaller the relative speed of the man needs to be ). However, it's still bugging me and making me believe the motion of the bottom plank affects the motion of the upper plank ( other than in the elaborated way ). Is there something wrong with this reasoning?

Note: Although this is not a school problem, I'll tag it as homework because this seems the kind of a problem that would appear in homework problems.

EDIT: By "stationary reference frame" I mean the frame on which the observer is standing ( e.g. Earth ), observing motion of the system"