How far will the boat move problem, help please?

[Marco]

Two people, one of mass 70kg and the other of mass 60kg , sit in a rowboat of mass 80kg . With the boat initially at rest, the two people, who have been sitting at opposite ends of the boat, 2.7m apart from each other, now exchange seats.

How far will the boat move?

Thanks

[Dirac]

The center of mass of the system does not move, as there are no external forces. Let d be the distance of the center of mass from a fixed point lying somewhere on the boat's fore-aft axis. Let A and B be the distance of the 70kg and 60kg passenger's seat respectively from this point. Let C be the distance of the boat's center of mass from the point. From the definition of center of mass we have, before the swap:

d × (70 + 60 + 80) = 70A + 60B + 80C

After the swap, the boat, and its seats, have moved a distance ?x, and the passengers are in the opposite seats, and d does not change (no external forces), so we get

d × (70 + 60 + 80) = 60(A+?x) + 70(B+?x) + 80(C+?x)

Therefore

70A + 60B + 80C = 60(A+?x) + 70(B+?x) + 80(C+?x)

?x = (A - B)/21

But A - B is the distance between the seats 2.7m, so

?x = 2.7/21 = 0.129m

The boat moves 0.129m