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This is the fourth of the five problems that

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Kortaggio sent me today.

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And I've been doing these problems because I think

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they're really neat applications of essentially

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fairly basic, even you could call it physics, rate problems,

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and a little bit of algebra.

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But you're able to figure out pretty neat things, where

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you're not sure if there's enough information at first.

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So, here we have train A, represented by these two dots

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and arrow, is 200 meters long.

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And I did this because I think we want to be specific about

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the configurations we're talking to later

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in the problems.

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Let's just read it.

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Train A is 200 meters long.

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Train B is 400 meters long.

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They run on parallel tracks at constant speeds.

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When moving in the same direction, A passes

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B in 15 seconds.

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From, so these are the configurations.

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From A is right behind B, to A is right in front of B.

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So let me draw that.

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So, train A, I'll do in this blue color.

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So train a is like that.

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And it's 200 meters long.

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And then train B, I'll do in this green color.

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Train B is double the length.

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So it's 400 meters.

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And so this is a starting configuration.

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The outline right here.

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And it says, it takes 15 seconds to pass it, to go

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to this configuration.

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So the ending configuration looks like this.

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Ending configuration.

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Train B here.

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And then train A has passed train B.

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200 meters.

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Once again, this is 400 meters here.

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And this situation takes 15 seconds.

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So this takes 15 seconds to happen.

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15 seconds.

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So, A passes B in 15 seconds.

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So this whole sentence right here, is this right here.

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We go from the situation to that situation.

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And then in opposite directions, they pass

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each other in 5 seconds.

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So from this configuration to that.

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So let me draw that.

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So, in the opposite direction example, they start like this.

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Train A is 200 meters.

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Train B is facing in the other direction.

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I'm doing it probably little bit too long, 400 meters.

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And then in 5 seconds they get from this configuration

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to this configuration.

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To that configuration, right there.

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200.

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And, of course, this right here is 400.

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And this takes-- this right here takes 5 seconds.

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So in opposite directions they pass each other in 5 seconds.

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Now, their question is how fast is each train moving

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in meters per second?

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So, once again they gave us-- they didn't really give us a

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lot of velocity information.

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They just tell us how long it takes to pass.

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But maybe using both of these pieces of information,

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we can solve it.

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So let's say that-- well, you have velocity

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of train A, so vA.

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Velocity of train A, and then of course you have

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the velocity of train B.

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And regardless of which direction they're facing, it

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assumes that they're always going at the same velocity.

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So in this situation, and we always have to just

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remember, distance is equal to rate times time.

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So relative, if we assume, because when you take a

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velocity, you can always take a velocity relative

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to something else.

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So, first of all, relative to this train right here, to train

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B, how far does train A travel?

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Well, it goes from this point.

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It goes from right here.

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To, not just 400 meters, it gets to the point where the

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front of the train is out here.

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So it has to travel 400 meters.

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And another 200 meters.

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So it travels 600 meters.

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It travels 600 meters in this situation.

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And how far does it travel in this situation?

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Well, once again, if we assume that this train is stationary,

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and that's, I guess, the key assumption we have to make.

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We're going to do everything relative.

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We're going to assume that, even though it is moving at a

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velocity, the position we're going to make relative.

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So, relative to this train, we move from right here,

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we move 400 meters.

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And then we move 200 more.

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So, again, in both situations, we move 600 meters.

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In this situation we move 600 meters, relative.

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I guess you could say, we move 600 meters relative to the back

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of train B in this situation.

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And we move 600 meters relative to the front of train

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B in this situation.

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In this situation, we do it in 15 seconds.

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That's because we're, kind of, where the velocity of train A

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is being eaten away by the velocity of B.

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If the velocity of B was 0, if this green train was really

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stationary, then we'd be moving relative to this train

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with velocity A.

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But now this is moving at some velocity.

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So our relative velocity to this train is going

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to be something lower.

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And what is the relative velocity?

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If you're a passenger sitting in train B, if you're a

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passenger sitting in train B, right there, how fast will it

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look like train A is going?

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What will be the relative velocity?

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Well, it's going to be the difference between the two.

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Right?

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So it's going to be vA minus vB.

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If you're sitting in this train right there.

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And you would say, it takes 15 seconds.

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So, rate times time.

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Times 15 seconds.

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Is equal to a distance that it traveled.

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And, once again, if you're a passenger sitting in this green

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train right here, you would say, OK, it went

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from this point.

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It crossed this entire train then it went

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another 200 meters.

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So it went 600 meters.

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Now, and this is of course in seconds.

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Now, obviously both of these trains in this have

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some positive velocity.

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This train would have moved even more than 600 meters.

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This train moved 100 and this train would've

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moved 700 meters.

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But I'm doing everything relative to what this passenger

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in the green train sees.

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Likewise.

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The passenger in the green train here, let's say the

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passenger in the green train right here.

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Oh, I'm doing his arms coming out of his head.

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But what velocity does he see this blue train coming in at?

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Well, he's going in this direction at 400

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meters per second.

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The other train is coming in-- no, sorry he's going in this

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direction at velocity of train B.

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We don't know what that is.

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400 is how long it is.

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And this train is coming with velocity, want to do it

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in blue, with velocity a.

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So you would add the two velocities.

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If this is coming at 60 miles per hour and this is going in

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that direction at 60 miles per hour, to this guy who's

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stationery in train B, he would just feel like this train

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is approaching him at 120 miles per hour.

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Or the addition of those two.

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So from this guy's point of view, this train is approaching

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with the velocity-- let me do, is approaching with the

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velocity vA plus vB.

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And in 5 seconds-- and they give us that information.

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In 5 seconds-- so velocity times time, or rate times

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time, is equal to distance-- it travels 600 meters.

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Remember, this is all relative to the guy, or the gal,

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sitting on this green train.

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And that's kind of the key assumption you have to make to

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make this problem solvable.

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Well, now we have two equations and two unknowns, we should

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be able to solve this.

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For the velocity of the two trains.

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So, just to simplify, let's divide both sides

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of this one by 15.

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So we have the velocity of A minus the velocity of B, is

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equal to 40 meters per second.

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Right?

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60 divided by 15 is 4.

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Yep, 40.

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And then here we have the velocity of A plus the velocity

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of B, that's an A, is equal to 120 meters per second.

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And, see, we could just take this equation.

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Put it down here.

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We could add the two equations to each other.

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So if the velocity of A minus the velocity of B is equal

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to 40 meters per second.

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Add the two equations.

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We get 2 times the velocity of A.

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These two cancel out.

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Is equal to 160 meters per second.

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Or the velocity of A is equal to 80 meters per second.

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And then we can just back-substitute here.

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The difference between the two is 40.

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So it's the 80 minus the velocity of B is equal to

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40 meters per second.

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So what's the velocity of B?

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Well, you could subtract 80 from both sides.

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You get minus velocity of B is equal to minus 40.

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Or the velocity of B is equal to 40 meters per second.

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And we've done the problem.

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And the key assumption there is to do everything relative to

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the green guy sitting inside of train B.

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You could have done it the other way.

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You could have picked other relative positions.

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But that, in my brain, is the easiest way to figure it out.

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So, this guy, in this case, is going at the speed at

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40 meters per second.

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And this guy is going at 80 meters per second.

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In this situation, this guy is traveling at 80 meters per

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second and this guy going in this direction at 40

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meters per second.

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Anyway.

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Thanks again to Kortaggio for that problem.

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