Person on a boat

Problem

Person on a boat. A person stands at one end of a small boat at rest on the water. They walk to the other end. Describe how the boat moves and why the centre of mass of the system (person + boat) stays put, neglecting friction with the water.

Solution

The person + boat system is horizontally isolated (friction with the water is negligible): the total momentum is zero and stays zero, so the centre of mass does not shift relative to the water.

For the CM to stay put, when the person moves forward in one direction the boat moves backward in the opposite direction, so that $\ev{m_p\,\Delta x_p + m_b\,\Delta x_b = 0}$ Since the person is lighter than the boat, their displacement is larger in magnitude, but the two mass-weighted contributions cancel exactly. In the figure the red point (CM) stays on the same vertical line before and after.

Links

Topics: Centre of mass Concepts: Centre of mass · Conservation of momentum