Boat with Friction: CoM / Minimum Time Problem

[quark002]

Homework Statement

A person of mass $m$ stands at the left end of a boat of mass $M$ and length $L$. Both the person and the boat are initially at rest. The coefficient of kinetic friction between the boat and the water is $\mu$. If the person starts moving to the right, what is the minimum amount of time for the person to reach the other end of the boat? Assume that the interaction between the person and the boat is continuous, as opposed to a discrete series of steps.

Homework Equations

$x_{cm_f} = x_{cm_i} + v_{cm_i}t + \frac{1}{2} a_{cm} t^2$
$v_{cm_f} = v_{cm_i} + a_{cm} t$
$F_{ext} = (m+M)a_{cm} = (m+M)g\mu$

Specifically, we have $v_{cm_i}=0$ and (placing the origin at the initial position of the person) $x_{cm_i} = \frac{L M}{2 (M+m)}$ and $x_{cm_f} = \frac{(L-d)m + (L/2-d)M}{m+M}$ (assuming that the person moves to the right and the boat consequently moves a distance $d$ to the left).

The Attempt at a Solution

I know how to solve for the CoM position as a function of time, but am stuck on how to get the minimum time. I'm guessing it would depend on how fast the person moves. It seems to me that there ought to be some constraint on the person's speed (in order for the problem to make sense), but I can't think of what it could be. Of course, I know that $(m+M)v_{cm} = mv_{person} + Mv_{boat}$, but I'm not sure how this helps. I also don't understand the significance of the continuity of the boat-person interaction.

 

[haruspex]

As you write, the question makes no sense without a constraint on the speed.
I also struggle to understand the "friction" model. It would mean that if the man accelerates slowly enough the boat does not move. Perhaps that's the missing constraint.