 This is called a graphing review. This is nothing to do in particular Tyson with this unit. I'm not teaching you anything about work and energy but on the provincial exam which no longer exists I realize but bear with me they would always on the written section second last question was asking you to interpret some kind of a graph and do something with it on your final exam which is an old provincial exam that I changed the numbers for and modified I guarantee you the second last question on the written section is going to be here's a graph interpret it on some of your tests in the future at least once or twice you're going to be given a question where I give you a graph and I say interpret this for me and many times in the big reviews that you're going to be getting so what I want to teach you are some clever Mr. Dewitt tricks of the trades methods that you can interpret graphs a lot of them memorized a bunch of crap last year in physics 11 probably most of you memorized that the slope of a distance time graph was velocity and the slope of a velocity time graph was acceleration it was a waste of your time you didn't need to memorize it I'll show you how you could have derived it in one second but I need to give you some background terminology first if we have two quantities two things that behave in the following manner and the first one is if y is zero when x is zero goes through zero zero and if x increases y increases so when x is getting bigger y is getting bigger first of all we say that they are directly proportional to each other directly proportional am I going to ask you to regurgitate that word no am I going to use that word and expect you know what I'm talking about probably or sometimes Alex will say y varies directly as x goes through zero zero and when x gets bigger y gets bigger and it's linear in a nice straight line that's kind of implied by this Courtney we write this y and this is the symbol for is directly proportional to x it's a little fish is the symbol for directly proportional to or if we want to turn this into an equation we write y equals little k x y little k because traditionally that's what they've used it stands for the word constant I thought constant began with a c not in every language only in English I think this was from German if I recall where the letter K is called the constant of proportionality the constant of proportionality again Courtney I'm going to ask you to know that word I'm not going to ask you to give it to me I might use that because I'm a math nerd this sounds really complicated it's not you've been doing this already without knowing that you were doing this keep forgetting to do that Mr. do it maybe I put it here did I note to self okay I'll just do it this way new page and let's even put some graph paper in the background here graph paper you've seen it before as follows if I gave you this the final is directly proportional to the time when time gets bigger velocity gets bigger don't write this down this is just an example to be a graph that looks like this if that was v and that was t and then I said to you Alex write the equation without realizing what you were doing you would have said oh the final equals a t it's v i plus a t but if we go through zero zero what's your initial velocity zero you were doing that physics 11 without realizing and in this case the constant of proportionality is the acceleration force is directly proportional to the mass can you complete the equation what has f and m in it what's the actual equation hello anyone okay f equals m a a is the constant of proportionality that relates force and mass or you could have said force is proportional to acceleration f equals m a m is the constant of proportionality that relates that relates force and acceleration you were doing it all the time without realizing it we just didn't call it that we called them equations nope wrong one here's an example the graph looks like this it goes through zero zero and it's a straight line goes through zero zero and it's a straight line there's all sorts of graphs in physics that do that what we want to ask you oh and the slope of this graph is k because it actually fits this y equals mx plus b but b the y intercept is zero so letter k is sitting where the letter m is it's actually the slope of the graph so we're going to apply it alexis when you're done yawning here's a graph d equals vt we're graphing distance versus time it asks what's the slope of this graph how do I find the slope of any graph I heard it here you're right rise over run let's write that down rise over run and let's assume this is supposed to go through ten comma five that's what I was trying to eyeball when I set this up really quickly alex the rise over run for a graph that goes through zero zero is really easy because that's your rise and that's your run what's the slope of this graph what is five over ten as a decimal now I want to ask you what does that mean and this is how I figure out what the slope of a graph means instead of memorizing a bunch of graphs alex what I do is I look at the units what's the y-axis measured in and that's the rise what's the x-axis measured in and that's a run and what I really get is this point five meters over seconds and what is meters over seconds a measure of we know it's velocity see if you memorize that last you're fine I never do I can derive it in one second by looking at the units this is hey velocity and Spencer I can figure that out for almost any graph that they throw at me I'll either divide the units or I'll divide the actual variables or I'll divide the actual variables for example if they gave me a work versus distance graph and they said what does the slope mean well what's work measured and first of all what is work measured in joules I would write that down sorry you guys just watch but I would write that down if I was doing this question what distance measured in so the first thing I would say slope is rise over run I would say well it's joules per meter do I have memorized a unit that's joules per meter no then I would rob move to the variables I would also say well what is the variable on the y-axis on the rise rob w over what is the variable on the x-axis rob I would say is there an equation that has w and d in it I would go looking at my formula sheet if I didn't have it in front of me Carissa not on her head Carissa what is the equation that has w and d in it so how could I make this look like this I think I would divide both sides by d with me oh you know the slope of this graph is actually a measure of force has to be Tyson you're right has to be so go ahead and memorize 80 graphs I think there's about 80 in physics 12 I don't you give me the slope I will either go rise over run and divide the units and try and figure it out that way my second resort is to go rise over one with the variables I could figure that out almost how do you throw it at me guarantee guarantee heck you can even do ones that you've never even seen before if I'm graphing momentum which is symbolized by the letter p and I'm graphing mass which is symbolized by the letter m can you look at your formula sheet right now and can you tell me what the heck the slope means haven't done momentum with you well let's see what's the equation that has a p m in it can you read it to me Spencer momentum equals mass times velocity so if I want to go momentum divided by mass Spencer what did you say the slope of this is I don't know momentum but I can figure that out okay so what do you do if it's a slope divide the units that next door is not someone's phone oh I was looking at it so for a second there hungrily thinking about 10 bits next page if you have a straight line but it doesn't go through 0 0 if y is not 0 when x is 0 but when x gets bigger y gets bigger we say there is should be an indirect variation they vary indirectly there's something else besides just the slope a y-intercept and we write y is proportional to kx plus b actually Alex the one that you're most familiar with is y equals mx plus b it's good old slope intercept form where m is the slope and b is the y-intercept so probably in physics 11 you would have somewhere along the line been given graph like this a velocity time graph a nice straight line and they would have said what's the equation well to find the equation we needed to first of all find the slope what's the slope of this what's the rise careful don't say the rise is 6 because it's not okay now this is one of the times in terms of bank my buck I almost always draw that because it takes me one second and it keeps me from making dumb mistakes and then I can say the rise is 3 over what's the run Spencer as a decimal it's 0.3 0.3 what what are the units well it's meters per second per second it's rise over run what is meters per second per second so if you memorize that last year fine but can you see you actually didn't need to because this is why Mr. Kamosy and myself last year emphasized to you hey memorize the units really helpful and alright what is that it's acceleration it's the acceleration what's the y intercept what comma what I'll give you a hint 0 comma Spencer now what does the y intercept mean on this graph though it means something here what is it yeah Cassidy what did we call that okay you know what I think that's vi on this graph isn't it over here let's try writing the actual equation and we're going to use our template y equals mx plus b except I'm not going to use the letter y what letter what variable is sitting on the y axis not a variable y what's sitting on the y axis v I agree equals m was the slope what did you say the slope of this line was 0.3 x oh no no I'm not going to use the letter x Shannon what variable sitting where the x is on my x axis no that's the units what variable what variable all I want is that right there you overcome what I'm going to use a t right there instead of an x yes yes yes that would make more sense plus keep going Shannon b is the y intercept what was the y intercept as a number by the way look up can you see v final equals a t plus v initial that's actually vf equals vi plus at in disguise it's where it comes from there it is hey it says right here what would the area underneath this graph b got your attention now now that you're breathing again I said the trick for slope is divide you know the trick for area is I don't know hey what's the area of a rectangle what's the equation for the area of a rectangle Rob what was the second word you said no second word you said second word you said not the first word not the third word the second word you said louder please what's the area of a triangle anybody know the area of a triangle Tyson what was the second word you said times you know what the trick when you're dealing with an area if you want to figure out what it means you don't divide you know what you do times here it says hey what would the area underneath this graph be I'm going to take my y units and times it by my x units or I would take my y variable and times it by my x variable because area is times one of these should help me figure out what's going on I think in this particular example it's easier to look at the units because I'm multiplying fractions math 12 how do I multiply fractions top times top bottom times how many s's on top Spencer how many s's on the bottom how many left Spencer what units are left what's the area underneath the velocity time graph what are you measuring meters and then it'll be fussy but if you said distance I wouldn't freak out I don't actually have memorized okay that's a fib I do have it memorized because I taught it so often but I didn't try to memorize that the area under a velocity versus time graph was distance I said I'll just multiply the units together hey here's graph what if I was graphing mg and h and you got something that looked like that and I said what's the area underneath here it's mgh this time I think it's easier not to look at the units I think this time it's easier to look at the actual equation and yeah Jessica just went well Mr. Deweyck says oh good gosh that's potential energy potential it's gotta be in fact I can even give you one that you may never have seen before now in theory you've done this in science 9 or 10 but let's see example C if when the x increases if when x gets bigger y decreases y gets smaller you get a graph that looks an awful lot like this quite often a curvy graph where math 12 I need to apologize to you because I believe physics has this wrong in physics in science in general if when one gets bigger the other one gets smaller we say y varies and they say inversely and I'm going to go on a rant here because actually it varies castity reciprocally inverses when you switch the x and y around and that's not what's happening here this is a case where I don't think the terminology is good but I'm stuck with what they use in science this is like to get inverse and reciprocal mixed up y equals k times 1 over x here's an example I'm graphing R in ohms this is resistance graphing I in amps this is current by the way what would the area underneath this graph be I don't know how would I figure it out area means I multiply the units I would go I times R can we look on their formula sheet and tell me what I times R is v which stands for not velocity that's a lower case v we're in the electricity unit what do you think it stands for? Voltage there you just discovered ohms law by using a bit of clever math sometimes you see this one y varies as x squared in fact you looked at it in math 11 it's a good old parabola d equals v i t plus I have a t squared oh kinetic energy equals a half m v squared there'll be a few more that show up this year kinetic energy equals a half m v squared Justin this particular parabola has a vertex at zero zero y because if you're not moving the kinetic energy do you have Justin this has to go through zero zero Justin you may remember from math 11 y equals a bracket x minus p all squared remember that bad boy so in this case a is half m what's the slope of this graph problem you can't oh who's a calculus okay in calculus you can find the slope are you guys starting derivatives yet derivative and it's not a very tough one but for the most part you can't what you would do instead Alex is you would straighten this curve out instead of graphing kinetic energy versus velocity from the page you could straighten this graph out by graphing kinetic energy versus velocity squared by the way what does the slope of this graph mean hmm let's try dividing the units that would be joules divided by meters squared per second by the way there's a typo here there should be a squared on this m right here because v squared will be m squared over s squared and on the previous graph we should scribble out the squared because velocity is meters per second not meters per second squared back to my point you guys know what joules over meters squared per second squared is I don't off the top of my head so instead of trying to divide the units I'm going to try dividing the variables this is kinetic energy divided by v squared do I have an equation that has kinetic energy and v squared in it which equation has kinetic energy and v squared in it let's write that down ke equals a half m v squared now Sabina let's make it look like kinetic energy over v squared what would I do with this v squared I think I would divide yes I think what we have is this kinetic energy over v squared which is your slope is oh you know what the slope of this is Andrew read this to me half the mass in other words if I'd given you numbers and said how heavy is the object find the slope and double it that's how heavy the object is if I'd given you numbers and said find the gravitational force on the object which might terrify some of you find the slope double it there's your mass times 9.8 there's the gravitational force all sorts of things you can start to do just don't panic relax slope divide the units or divide the variables area multiply the units multiply the variables and in fact that's the lovely summary to do the slope divide the units or the variables to do the area multiply the units or the variables and 99.9 percent of time that'll get you there what about that 0.1 be clever there'll be something else that gives it away is that okay Kehishan what to say here a week from this Thursday what's it say keep reading Kehishan we are done the unit what's your homework if you haven't handed in lessons one or two or three get them in otherwise it's going to be take home quiz work on the review before I turn you completely loose bear with me for a moment