Boat

One is usually remembered that under such conditions the center of gravity (of the BOAT + MAN system) remains in place. And - after drawing a little - we got equation for the boat shift/offset: 

Here L is length of the boat, m is mass of man, M is mass of boat.

Or - let say - "from first principles": 

Those who remember only Newton's laws write that (in the inertial frame of reference associated with the shore) the product of the boat's mass M by its acceleration A is equal to the sum of all forces acting on the boat. And only the person (of mass m) who walks along the boat with acceleration a acts on it: 

Integrating twice in time (from the beginning of the person's movement to his stopping, or to infinity, when everything stops), we get the products of masses and displacements (of a boat and a person) relative to the coast . 

And since the displacement of a person relative to the coast is: 

из (3) снова получаем (1)! 

But suddenly ... 

a clever man appears and said:

- What if we take into account the friction on water? 

Well! Recall that for low velocities, the fluid friction force is proportional to the body's velocity V , and the proportionality coefficient k depends on the properties of the fluid, the size and shape of the body.

Now - according to Newton's laws - the product of the boat's mass M by its acceleration A is equal to the sum of two forces: from the person walking on the boat, and the force of liquid friction. Instead of (2) we have:

Integrating over time (from the beginning of a person's movement t = 0 to infinity, when everything stops), we get: 

But all velocities - both the boat and the person - are equal to zero at t = 0 and at infinite time (when everything stops). Thus, for any k other than zero, the boat displacement is: