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unit analysis and geographical data analysis who spilled all these letters

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in my math unit analysis make your leaders and meters work together...[mumbling]

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Alright let's get started hope you're in the mood for some algebra [Ring beaming colors]

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remember the Pythagorean theorem a squared plus B squared equals C squared

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looks pretty algebraic to us and when we're dealing with formulas in physics

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we've got to be able to solve for the different variables we'll be using

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so we'll need to isolate those variables and make sure we're using the right [Variables in a prison cell]

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units at the same time and using the right units it is important we're going

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to be dealing with more than just meters and seconds we'll be looking at things

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like kilograms times meters per second squared and sometimes we're going to be

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dealing with equations where we just want to see how the unit's involved

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relate with each other without plugging in any specific numbers which is called

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unit analysis we'll put unit analysis to work when we need to solve for a

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particular variable and once we've isolated that variable that's when we [Hand picks up E variable]

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can plug in any numbers that we have so how do we isolate a variable well the

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laws of algebra say that anything we do to one side of an equation we do to the

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other side as well algebra is all about equality so if [Man and woman sitting on a seesaw]

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we've got an equation that says A plus B C over D and we want to solve for A then

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we have to subtract B from both sides meaning A equals C over D minus B and

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when we're doing algebra or any math really we can't forget the order of [PEMDAS appears with corresponding symbols]

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operations and we can't forget that when we've got PEMDAS lodged into our brain

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yep another nonsense math word PEMDAS tells us what to do first when we're

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dealing with a complicated piece of math it's also called the order of operations

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so we solve anything in parentheses first then we handle the exponents after

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that comes multiplication and division and really those two have equal standing

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in the land of PEMDAS so if you have a multiplication and division in a problem

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just solve them going left to right and bringing up the rear are addition and

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subtraction those two are also equivalent so do the left to right thing

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with them too okay okay enough talk let's see how we apply this stuff to a [Woman discussing unit analysis]

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physics problem so you know how there's the Sun and then there's the earth where

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you know we're all just chillin and you know how they're really far away from [Earth and Sun in space and boy appears scratching head]

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each other let's play around with stuff to find out just how far apart they are

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and as you might expect we'll start with gravity the force of

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gravity between two masses like the Sun and the earth is given by the equation F

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sub G for the force of gravity equals G which is the gravitational constant

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times the mass of the first body M sub one times the mass of the second body M

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sub 2 divided by the square of the radius R whoo that made us a little dizzy

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what do we do if we want to solve for R let's start playing with the equation [Person holding phone and formulas appear]

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first off let's multiply both sides by r-squared algebra is always fair like

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that both sides of an equation get equal treatment now our equation looks like

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this r-squared times F sub G equals the product of G M sub 1 and M sub 2 now we

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have to get r-squared all by itself so to isolate R squared we

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divide both sides by F sub G like this leaving us with R squared equals G M sub

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1 and M sub 2 over F sub G but wait we're trying to solve for R, not R [R and R squared waving]

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squared so one more step if R squared equals let's just say everything on the

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right we're tired of repeating all these letters sticking to the zen of algebra

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we need to find the square root of each side to drill all the way down to R [A nail drilled through piece of dry wall]

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since the square root of R squared is R that means R equals the square root of G

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M sub 1 M sub 2 over F sub G boom! we've got R all by its little lonesome ready

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to be solved now did you notice how we kind of breezed by G in this equation [Woman with arms wide in the breeze]

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just dropped a gravitational constant in your lap and kept on moving well let's

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take a closer look at what makes up that constant we're not going to find the

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actual number for that constant we're just going to figure out what units it's

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made of so we'll start back with F sub G equals G M sub 1 M sub 2 over R

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squared multiply each side by R squared then divide each side by the masses M

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sub 1 and M sub 2 so G equals F sub G times R squared over M sub 1 times M sub

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but now we need to substitute units in

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place of the variables that aren't G ...Force is measured in Newtons so F sub G

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gets replaced with N... A radius is a length so that gets replaced with meters

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and those meters are still squared then on the bottom we're dealing with masses

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which means kilograms since it's kilos times kilos that means it's yup kilogram [Captain America appears below force equation]

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squared so the units that make up the gravitational constant are newtons times

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square meters / square kilograms not quite as simple as our old friend the [Meter standing on a chair]

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meter now we can put everything together use some numbers and get some real

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answers let's take one more look at the gravity equation we've been working with

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yeah we're totally not tired of looking at it either but this time we're going

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to use actual numbers and we're going to figure out the distance between the Sun [Distance between sun and earth appears]

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and the earth and we're sure you remember we already arranged the

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equation to solve for R... R equals the sub square root of G, M sub 1 and M sub

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2 over F sub G time for those numbers we mentioned the gravitational constant is

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6.67 times 10 to the negative 11th Newton's meters squared

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over a kilogram squared see why we did that unit analysis just now Earth's mass [Earth appears in space]

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is 5.98 to the 24th order of magnitude kilograms and the Sun

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has a mass of 2 to the 30th order of magnitude kilograms

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Wow the Sun is big and last but not least the force between the Sun and the

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earth is 3.55 times 10 to the 22nd Newton when we put all those

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into our equation for R it looks like this now we can start figuring this mess

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out activate PEMDAS while we do have parentheses and exponents here they are [PEMDAS appears under an equation]

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keeping the unit's clear and easy to understand so first off let's multiply

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the numerator and while we're at it we can cancel out the kilograms since our

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gravitational constant uses the inverse of kilograms squared

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and our masses each used kilograms we're left with 7.98 times 10 to the 44th

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Newton meters squared over 3.55 times 10 to the 22nd Newtons

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oh yeah and we're still looking for the square root looks like it's time for

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division and again we're able to cancel out a unit in this case Newton's after [Newtons cancelled out from formula]

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we divide we find that R equals the square root of 2.25 times

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10 to the 22nd meters squared doing our last bit of math tells us that the earth

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is 1.5 times 10 to the 11th meters from the Sun or to put it a little less

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scientifically really really far away so we've explored how to work with data [Independent and dependent variables in a table appear]

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collecting it using the right significant figures for it making sure

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we're using the right units for it all sorts of fun stuff but wouldn't it be

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nice if we could actually look at our data really be able to see the pattern [Man examining data with magnifying glass]

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of how changing an independent variable affects the dependent variable good news

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graph paper exists graphing data lets us see how variables are related for now

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we're going to just use a two-dimensional graph meaning we're [Graph appears]

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going with the classic x and y coordinates when we're using a graph to

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represent data from an experiment like the pendulum experiment we did the [Man and woman with swinging pendulum]

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independent variable will be graphed horizontally on the x axis and if you

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guessed that the dependent variable get graphed vertically on the y axis then

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yeah it was the only option left but hey it was still a good guess let's do a

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little refresher on some different kinds of mathematical relationships sometimes

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we get lucky and have a nice pretty straight line something easy to deal [Straight line drawn through an axis]

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with and in those cases we're going to pay attention to the line slope the

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slope can be as steep or shallow it can be negative or positive and I could also

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just be flat if going from left to right that line goes up then the slope is [Slope appears on graph]

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positive if it goes down the slope is negative and if it's flat yep

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that means the slope is zero and just in case someone wants to

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get tricky on a test if you ever see a graph like this with a straight vertical [Graph with straight vertical lines]

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line that means the slope is undefined which occurs when something is just

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trying to mess with our heads or when we screwed up something in our math hey

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everyone makes mistakes slope is rise over run

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meaning rise meaning the y-axis and run meaning the x axis to put it more

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mathematically the slope M equals the Delta of Y over the Delta of X in this

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context Delta represents the change in value so the slope equals the change in

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Y divided by the change in X if we have a steep slope of Mount Everest slope, that

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means there's a big change in the value of y relative to the value of X and any

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slope can be discovered using the linear equation which says that y equals M [Linear equation appears]

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times X plus B and represents the slope and B represents the y-intercept which

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is where the line crosses the y axis let's look at an example first what's

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the slope of this line well we just need to look at any two points along the line

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let's use the point at 3,10 and the point at 5,16 using our equation for slope we

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find that M equals 16 minus 10 divided by F minus 3 [Equation for M appears]

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making the slope equal to 3 so using that slope in the linear equation along

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with the y-intercept value of 1 we can describe this graph using the linear

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equation y equals 3x plus 1 of course not all lines are straight so that'd be

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too easy we also have curved lines now there are all sorts of curved lines and [Person drawing curved lines on graph paper]

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all sorts of nonlinear equations to go along with them but we're gonna focus on

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three the first curved line we'll look at is the quadratic or squared line

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it has a parabolic shape like this and the equation that describes it is y

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equals AX squared plus BX plus C in this equation A, B and C are constants and you

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may not need A or B if they're equal to 1 a quadratic line can have up to 2

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roots and in this context a root is an intercept where the x value crosses the

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x axis where the y equals 0 you can see these here at negative 1,0 and 2,0 the

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next line up to bat is the rational line and the matching equation is y [Rational line appears]

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equals A over X minus B you might also see this called an inverse function and

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there's also a squared inverse that's y equals A over x squared minus B lastly

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for this lesson is the radical line it's called a radical because the function is [Radical line appears on graph]

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under a radical sign, not because it's really into surfing and because it uses

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a square root it's also called a square root function and the equation is y

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equals the square root of AX minus B and the line looks like a sideways parabola

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see if we want to know the precise slope at a precise point on a curve we can get [Tangent curve appears]

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it by finding the slope of a line that is tangent to the curve of that point it

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would be great if we could see all of this graphing stuff in action if only we

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had some data to put on a graph let's say like say we've done an experiment [Newton's cradle appears]

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maybe with a pendulum and we'd written down a bunch of stuff measuring changes

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to variables sound familiar? remember the independent variable is on

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the x axis and the dependent variable is on the y axis [Table of dependent and independent variables]

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so go ahead start to plot some lines on a graph see if you have a straight line

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or a curve and if you have a flat line with a zero slope that means the

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dependent variable isn't related to the independent one so changing the

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independent variable doesn't affect the dependent so go ahead get plotting see [Person plots lines on a graph]

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what direction the data points to have fun drawing your lines just don't get

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too carried away if your graph ends up looking like a van Gogh painting maybe

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save that one for art class instead of physics

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