You're with 19 other people on a boat at rest in frictionless water. The group's total mass is 1

You're with 19 other people on a boat at rest in...

Key Concepts

Relative Motion in Isolated Systems

In the absence of friction and external forces, the relative motion of the individuals with respect to the boat causes an equal and opposite reaction on the boat. Understanding relative motion helps explain how the movement of people affects the boat and leads to its displacement even though the total momentum of the system remains zero.

Mass Distribution and Its Effect on Movement

The difference in mass between the boat and its passengers plays a key role in determining the distance each moves. Since the boat has a much larger mass, a small shift in the center of mass of the passengers results in only a small displacement of the boat. This concept is important for understanding how the redistribution of mass within a system impacts the movement of its larger or more massive components.

Conservation of Momentum and Center of Mass

In an isolated system with no external forces, the overall center of mass remains constant. When people move inside the boat, the boat must move in the opposite direction to keep the center of mass fixed. This principle is a specific application of the conservation of momentum and is critical in understanding how internal movements affect the motion of a system as a whole.

Transcript

00:01 Okay, so in this problem, we're 19 people at rest in frictionless water.

00:06 Okay.

00:07 And then the total mass is 1 ,500.

00:09 The boat's mass 12 ,000 kilograms.

00:13 And we walk from boat to stern.

00:17 That sounds fun.

00:19 And we want to get how far does the boat move? okay, this is interesting.

00:24 So let's just get down all the givens.

00:25 This reminds me of a problem that i got wrong on a test once.

00:28 So hopefully i can conquer it here.

00:32 So 19 people.

00:36 So let's just do n is 19.

00:40 So mass of people, let's just say that's 1 ,500 kilograms.

00:51 And the boat's mass is 12 ,000.

01:04 And the distance is 6 .5 meters.

01:13 And we want to get, so that would be like the change in x of the people.

01:19 So we want to get the change in x of the boat.

01:21 So what we want to use is momentum conservation, kind of.

01:28 I guess it would be more like, so you can say f is the change in momentum over change in time.

01:37 If there's no outside force, then the change in momentum has to be zero.

01:41 So that's m, d, v of the center of mass, dt.

01:52 And so that if the velocity isn't changing, the velocity has to be constant.

01:57 And if it starts out in zero, then the position, if the velocity of the center of mass starts out zero, then it must remain at zero.

02:08 Therefore, the center of mass isn't moving.

02:11 And therefore, you can get that, the important conclusion that the center of mass doesn't change.

02:18 It's such a simple principle, and i remember really overthinking it.

02:23 That's the main idea that you want to get.

02:27 Now, if the boat, i guess we went into a frame from an onlooker of the boat, then it would change it.

02:34 But in the frame of the boat, the center of mass is not changing.

02:37 So what we need to do is calculate the center of mass before or after the people move.

02:43 So let's draw a nice little picture in this little area.

02:49 Let's put this all to the side.

02:51 So here's the boat.

02:55 And this dot is the people.

02:57 And then they're going to go over here.

02:59 And then that means the boat has to kind of go to the left to keep the center of mass the same.

03:06 So let's calculate.

03:08 Well, originally actually, yeah, let's come up with a formula for the center of mass before and after.

03:13 So center of mass before...

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