Two people, one of mass 85 kg and the other of mass 55 kg, sit in a rowboat of mass 58 kg. With the boat initially at rest, the two people, who have been sitting at opposite ends of the boat, 3.0 m apart from each other, now exchange seats. How far and in what direction will the boat move?

VIDEO TRANSCRIPT

This is Giancoli Answers with Mr. Dychko. We have these two people sitting at each end of the boat and they are gonna switch places and the question is how will the boat move when they switch places? Well, first of all, you need to understand why the boat's gonna move and the reason is that the center of mass has to stay in the same place even when these guys are moving around because there is no external force on the system so the center of mass won't move; there's no net force. Now yeah sure there's a force that the person's applying to the boat and which the boat also applies to the person as they walk but those two forces are internal to the system so they don't move the center of mass. So let's figure out where the center of mass is to begin with with their original positions and then when the people switch positions, we'll say the center of mass has to be in the same place and well where does the boat have to move in order to make sure the center of mass stays there? So center of mass to begin with is the mass of person A times their distance to the origin which we are gonna choose to be at A's original position so this x A's gonna be 0 plus mass of the boat times its distance to the origin plus mass of person C times their distance to the origin divided by the total mass. And so that's 58 kilograms, mass of the boat times well, assuming the boat is of uniform construction so that there's even distribution of mass throughout the length of the boat that means the center of mass of the boat will be in its geometric center so half of three meters and then plus 55 kilograms mass of the person times their distance of 3 meters from the origin divided by the total mass of 85 plus 58 plus 55 gives 1.27 meters is the position of the center of mass of this system before the people switch positions. Now when they switch places, we need to have the center of mass still located there— that's the rule because there's no external net force applied to the system so the center of mass will remain where it is— but now with the positions of these people being given to us, we are told that they switch places and they go at opposite ends of the boat; the only thing that we can change such that the center of mass stays here is to have the boat move and how much does it move is the question. Well, let's suppose the boat moves to the right; we are gonna find out that it actually moves to the left but doesn't really matter what assumption you make, I mean we are gonna get an answer that's negative which tells us that this picture is not quite right, in fact the original position is gonna be here instead of over there but that's fine. So the position of the center of mass is same formula as before—up here— except none of them are gonna disappear now because none of these center of masses are positioned at the origin anymore; we have person C who is now at the left end of the boat is gonna have some distance x C from the origin and x C is what we need to find in this question— it's the distance that the boat has moved— plus the mass of the boat times its distance from the origin, x B, plus mass of person A times their distance from the origin and then divided by the total mass. So m Cx C, can't do anything with that but then x B we can write as half the length of the boat plus the distance x C. And then person A is the length of the boat 3 meters plus this distance x C and we are gonna do some algebra to solve for x C. So we'll get rid of these brackets here by multiplying m B by 3 over 2 which is 1.5 and then m B times x C and then 3 times m A and m A times x C as well. And then multiply both sides by this denominator m A plus m B plus m C so that makes x center of mass times m A plus m B plus m C on one side of the equation and we are switching the sides around as well so that we have x C on the left. This x C is a common factor among these three terms here and so it's x C times m A plus m B plus m C plus 3 times m A plus 1.5 times m B and all that equals the center of mass position times the total mass. And then subtract the 3m A from both sides and subtract 1.5m B from both sides and so that makes those terms go on the right hand side with a minus there and then also, after doing that, divide by this m A plus m B plus m C to solve for x C and then plug in numbers. So we have 1.2727 meters—position of the center of mass— times the total mass 85 plus 58 plus 55 kilograms minus 3 times 85 kilograms minus 1.5 times 58 kilograms and then divide by the total mass and that gives negative 0.45 meters. So that's 0.45 meters toward the initial position of the 85 kg person and we kinda have to say it that way because the question doesn't specify who was sitting on the left to begin with; whether the 55 kilogram person was here first or the 85 kilogram person. So to be totally unambiguous, maybe a better way of saying this is to say that the boat moves 0.45 meters toward the initial position of the 85 kilogram person. And there we go.