ShmoopTube

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No grafs tell stories and stories tell us equations Okay

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using graphs to tell a story that's Pleasant feeling that's

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What stories Dad in the matter A lot of math

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Okay Equations instead Action movie equation Yeah Okay who didn't

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love story time when they were a kid Who indeed

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you'd gather around have your mom or dad or your

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teacher read from a picture book and they'd hold up

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the pages so you could see the illustrations everyone would

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Who and ah and physics is just like that Okay

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there may be a few small differences but think of

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grafts They tell us stories through pictures We just need

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to know how to read them If you've been following

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along with our course so far you've already seen a

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bunch of displacement versus time Graphs like this one Displacement

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versus time gives us information about velocity versus time And

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in the same way velocity versus time tells us about

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the acceleration versus time But rather than just yammer on

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about it let's see what story this graph is telling

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us What do we see here We see something that

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starts moving picks up speed then slows down What story

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could we think of that fits this Sure there are

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a lot of situations to choose from That would match

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this like a car going from one stop Sign to

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another Although the car would probably be going faster than

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this What if uncle frank decides to relive his roller

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disco glory years The seventies were a crazy groovy time

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Now frank isn't the king of the roller rink like

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he used to be So when he comes to a

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hill he's just gonna roll with it literally he's not

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going to add any force He's just going to pray

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he doesn't fracture his pelvis We'll be observing him helping

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the video of disco uncle goes viral so we'll say

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frank is rolling in the positive direction away from us

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He rolls down the hill for about two point five

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meters when he starts going up the next hill which

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means he slows down of course and he stops after

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ten seconds five meters away from the us Sure that

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story where it's great for those graph Maybe not so

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great for frank though And what does this graph tell

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us about velocity over time Well since the overall slope

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of the line is upwards the velocity will be positive

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And since this is a curved line the velocity isn't

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constant It changes throughout the time period So if we

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were to plot out a velocity versus time graph It

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would look like this Frankie starts out at zero meters

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per second preaches a max of one meter per second

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and then gravity kicks in and slows him down until

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it's back teo zero meters per second Before we go

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onto acceleration let's look at the slope of the velocity

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graph more carefully And let's Think about what it means

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for acceleration We can see that the slope is constant

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and positive from zero to five seconds then abruptly changes

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to constant and negative in the five to ten second

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region That abrupt change of five seconds tells us that

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is the moment when the acceleration was not uniform In

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other words acceleration changes at five seconds as we're sure

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you mutter in your sleep these days acceleration is the

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change in velocity divided by the change in time Yep

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That equation looks pretty familiar So for the first part

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of the velocity graph the outboard part we would figure

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out the acceleration like this Well look at the change

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in velocity from second zero two second four Why aren't

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we going all the way to the top two second

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five Because we already know that the acceleration at that

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time is zero that's the transition point So in that

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four second time span the velocity goes from zero meters

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per second Two zero point eight meters per second We

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divide that change in velocity by that change in time

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to find that our acceleration for the first four seconds

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And when we do the math we find that the

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acceleration is a constant point Two meters per second squared

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let's Slap that sucker on the acceleration versus time graph

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Let's look at the other part That part has the

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downward slope A second six through ten Another four seconds

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Man at second ten our velocity is zero meters per

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second at second Six it's point eight meters per second

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giving us a change of negative point Eight meters per

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second Divide that by the change in time which gives

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us an acceleration of negative point Two meters per second

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squared Well pop that on the graft too So right

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now our graph looks a little dysfunctional huh A little

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disconnected like us before our first cup of coffee So

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let's fix that by adding the point at the five

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second mark where The acceleration is zero Then we can

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connect our two horizontal lines much better Looking at that

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disconnected graff was making us itch It's a super important

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to keep the time when the acceleration is changing out

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of both the before and after pictures Think of what

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would happen if we looked at zero through five then

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five through ten then at the five second point we'd

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have a vertical line Ah vertical line on a time

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graph is bad Very very bad acceleration is change in

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velocity over change in time if there's no change in

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time that means we'd be dividing by zero And as

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we were all taught in math class when you divide

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by zero you open a portal to hell that swallows

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the universe in a fury apocalypse Ok it's not quite

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that bad but she can't divide something by nothing It's

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nonsensical and in math terms is considered undefined And if

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you have a vertical line on an acceleration vs the

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time graph that would mean that the acceleration was infinite

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Which just know nope Uh uh not gonna happen So

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graph carefully when acceleration is changing So displacement velocity and

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acceleration are all related part of one big happy family

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let's take a look at some ways to work with

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them Time for some equations We've already looked at some

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basic equations Displacement equals final position minus the original position

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Velocity equals the change in displacement over a period of

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time Acceleration is the change in velocity over a change

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in time But how about this bad boy What the

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heck is is trying to say let's Break it down

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x final is our final displacement wherever we've ended up

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in other words x sub zero is our initial displacement

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also known as where we started out these sub zero

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is our initial velocity and t stands for time so

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this is initial velocity multiplied by the elapsed time a

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is acceleration pretty standard Auntie is still time But in

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this part we square it so putting it all together

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our final displacement equals our initial displacement plus our initial

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velocity times the time period plus one half acceleration times

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time squared Now you may look at this See all

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those variables and want to crawl under your desk and

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die We get it But all those variables are a

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good thing because it means that if we have enough

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info we can solve for the initial velocity initial or

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final displacement time or acceleration Look at that versatility That's

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like the swiss army knife of equations How about this

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one This says that the final displacement minus the initial

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displacement equals the final velocity minus the starting velocity divided

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by two multiplied by the elapsed time So to put

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it another way the change in displacement equals the average

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velocity multiplied by the elapsed time so we can know

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how far we went If we have our average velocity

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and a clock that could definitely come in handy But

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what if we don't have the time Well we always

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have the time for math But what if no one

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was watching the clock So we don't know how much

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time passed Good news there's An equation we can use

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in that situation Here it is Let's Break this one

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down The final velocity squared equals the initial velocity also

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squared two times acceleration times the difference between the final

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displacement and the initial displacement which is also known as

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the change in displacement This cuts the time right out

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and lets us solve for the initial or final displacement

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initial or final velocity and acceleration How do we know

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which equation to use We take attendance that's How Just

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like in class for these kinds of questions we've only

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got six variables that we're going to run into We've

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got the displacement brothers that's initial and final displacement and

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the velocity twins initial and final there too Then we've

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got acceleration and last but not least time no relation

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between those two look through the question see which variables

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were given and which we need to solve for So

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if both displacements both velocities and acceleration are all present

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and time isn't mentioned we'll pick the right equation for

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the job and sometimes we might need more than one

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equation like if there are different kinds of motion that

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the car is at a stop sign accelerates then cruises

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on the freeway for a while If we want to

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know the total displacement we'll need an equation that describes

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this displacement during the acceleration un equation that tells us

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the final velocity and the one that tells us how

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far the car went at that velocity or what if

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we have a case of two moving objects like the

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old classic of two trains if we have a train

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going from chicago to detroit and another going from detroit

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to chicago will need different equations for each to figure

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out when they're going to crash Come on just having

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them past each other is so boring So how does

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this whole taking attendance thing work Exactly Let's see it

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in action say we've got a superhero situation since we're

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always staying on brand let's call her shmoop er woman

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she's just chillin up on a roof when she hears

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a distress call she takes off running thirty meters to

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the end of the roof She hits the edge of

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the roof and takes flight When she gets to the

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end she has a velocity of five meters per second

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How long does this sprint taker to cover those thirty

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meters time for attendance Unnatural displacement here there mary is

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And just because he's zero doesn't mean he's worthless What

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about the final displacement Yep Thirty meters right Their initial

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velocity Bingo Zero meters per second Final velocity That's definitely

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showed up five meters per second acceleration Uh acceleration Anyone

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sane acceleration Uh yes acceleration is out sick How about

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time Oh yeah That's what we're trying to figure out

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so which equation should be used Let's look at some

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of our options since acceleration isn't involved we don't want

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any equation with an a so we can scratch the

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acceleration equation off the list and we can also scratch

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captain versatile off also the big one with all the

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variables we could use it but we'd have to calculate

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acceleration first So let's look at something simpler This one

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requires us to find the average velocity first again Sounds

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like extra work and we know we don't want this

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velocity equation It's whole thing is that it doesn't involve

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time That leaves us with this equation Remember this one

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final displacement minus initial displacement equals final velocity minus initial

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velocity divided by two times the time that's got everything

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we need Okay first of all let's isolate t that

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is what we're trying to solve After all let's get

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algebraic on It will start by multiplying both sides by

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two so now we've got two times the difference in

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displacement equals the difference in velocity times time one more

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step to get t all by itself divide both sides

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by the change in velocity Leaving us with this time

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equals two times the difference in displacement over the difference

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in velocity Let's start putting in the Numbers the final

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displacement was 30 meters and the initial displacement was zero

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meters so those go there and the final velocity was

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five meters per second with an initial velocity of zero

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meters per second So we'll pop that in right there

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we won't actually do the whole subtracting zero from numbers

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thing we think he can handle that just fine so

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we have sixty meters on top and five meters per

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second on the bottom sixty divided by five is twelve

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and when we look at the units the meter's cancel

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out leaving us with a time of twelve seconds and

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shmoop er woman saves the day again A lot of

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math is like telling a story we've got your beginning

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when you get all the numbers and what to solve

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for then you've got your middle where you make any

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rearrangements and do the math and at the end you

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got your answer but will grant you that meth is

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in the most Popular story to tell You'll probably see

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a superhero movie in the next few months But equations

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Those aren't quite ready for the big screen yet Bring 00:13:50.818 --> [endTime] it back

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