hello and welcome to the final part of chapter two the final lecture covering chapter two this is the fourth of four lectures that introduces motion okay the study of motion known as kinematics which is chapter two of the physics of everyday phenomenon they are which is what this lecture series is all about so we've been talking about acceleration we led up to it with a discussion of speed and velocity okay all the study of kinematics now we finished with defining acceleration we know it's a vector we know they can point in a certain way in particular situations we know that it can change when both the magnitude or the direction of velocity is changing or both is changing okay so all that is wonderful in terms of a general definition but what we're going to have here is a workhorse of a particular situation which is when the acceleration is constant so acceleration is just going to be some fixed value like 9.8 meters per second squared mentioned that a couple times right we'll see that more in the next chapter too so it's just going to stay at that fixed value okay so with that in mind we have a whole set of ways we can quantify that that affect that that resulting motion okay all right so uniform acceleration is the simplest form of acceleration but still a really important one okay so it's not like we're actually missing the point here it occurs when there is a constant force acting on an object okay that could be like a constant tension force it could be a constant buoyant force it could be a constant gravitational force uh as we get into other types of physics it could be a constant electrostatic force or a magnetic force but if it's constant which means it doesn't change with distance doesn't change over time okay then we have constant acceleration constant force constant acceleration remember that okay so most of the examples we consider will involve constant acceleration right it really is ubiquitous it's very common falling rock okay car accelerating right so you know passing someone you assume they're they're going to be accelerating at a constant rate which is you know pretty accurate because you kind of like you you push you know down the pedal a certain amount you push that same amount throughout your moment of acceleration constant acceleration is pretty commonplace okay so the acceleration does not change as the motion proceeds that's what we mean by uniform acceleration or as i keep calling it constant acceleration now i think your book is a little loosey-goosey about this which is fine but usually uniform is the same in space like x and y right or z if it's three-dimensional and constant is the same in time okay now here we actually want the acceleration to be both uniform and constant and those terms can certainly be used interchangeably the point is it does just doesn't change okay throughout the system the situation that we're describing the word problem no change okay the example whatever okay so for example a car moving along a straight road and speeding up at a constant rate right so just like you know maybe the pedal all the way down so just that constant rate of acceleration would have a constant acceleration so if we graph the acceleration this is actually the second time we've seen a graph of acceleration both of them become weird the last in the last video we saw little spikes showing you know sudden changes between velocity okay sort of disruptions if you will here it's just a flat line because it's just a steady value okay this is that constant force so notice that even though the acceleration is not changing the speed will be changing right because you are you know you're you're speeding up constant acceleration is still a change it's a change in speed okay we'll see it's a linear change in speed and it will create a square change in position so we're actually going to end up with a parabolic displacement okay a parabolic position function so in that case of the car accelerating at a constant rate our velocity would increase at a linear rate okay because remember acceleration is just the change in velocity over time okay therefore the change in velocity okay so the delta v here is just going to be equal to acceleration times time okay so think about that now that's great right delta v you know if if you had a general statement because you could just say oh well you know this is my change in v it's equal to a times t and then you do too much with that if if the acceleration itself is a function of time okay for those of you that that know calculus and it's fine if you don't then you end up with a situation where you have acceleration as a function of time multiplied by the variable right the only way to deal with that is the the mathematics of calculus but in the special case where the value a right the acceleration value is not of a a function but is instead just a constant it's just a number well what you get is the equation of a line because what is delta v after all delta v is v final minus v initial and we can sort of rewrite this in a way to make it look like a particular equation sort of make it look like the equation of a line okay so let me go over to here we go where i will show you what i mean okay because what we have is that delta v by definition is the change in something the way we've seen this before is we've seen delta v just equal v final minus v initial okay that's perfectly acceptable notation right f for final i for i for initial using subscripts we do that a lot with these labeling of things physics is a lot of labeling okay just like in math but here's you know it's a physical system right so now math is so well suited to physics it's the science that really just needs the most of that of all these different variables okay now here's the thing though that we want to instead refer to v final as sort of the instantaneous velocity it's the function right so instead we'll write delta v as just v right whatever is the current value of velocity and then to give sort of special attention to the initial value we'll call that v naught okay so here the zero is kind of serving the same role as the i for initial it's zero for not which just means the initial okay so when then we if we were to rearrange that right because all i've done is change the left hand side of the equation replace it with v minus v naught right which by definition is delta v that i can just add the v naught the initial velocity of both sides and now i have a velocity function okay so this is a linear function for velocity specific to the very common important situation of constant acceleration that's what we're talking about okay so this is a a particular function that you can use if you know the value of acceleration like the car accelerating or something in free fall right because in free fall and we'll see this in the next lecture as well that value of constant acceleration is none other than the value called little g which is 9.80 meters per second squared okay that's kind of the global average it actually does vary into that the um the hundreds place of the of the meters per second squared okay depending on kind of what continent you're on different density constants have slightly different gravitational acceleration it's kind of cool idea but we'll talk about that when we talk about where gravity comes from but the point is that the global average is taken is either 9.81 or 9.80 depending on who you ask i think 9.80 or 9.80 is a good value to use okay now you will also see gravitational acceleration sometimes rounded to 10. i'll always be clear on whether i want you to round it or not i usually will not if you're in my class okay i think it's perfectly fine to not round it there's only some cases where you might want to because the powers of 10 can be really clean for making a good argument okay which is a good example now getting back to velocity though in this idea of linear velocity no matter what the actual numerical value of acceleration the function for velocity as a function of time okay for this this case of uniform or constant acceleration is a diagonal line because it is the equation of a line okay now in this case when this diagonal line passes through the origin right the intersection of the vertical and horizontal axis the origin well that happens that's a common case but actually if you look at the equation it only passes to the origin when what is equal to what okay i'm covering up part of it when v naught is equal to v well excuse me zero okay so this is only true when the initial velocity is equal to 0 meters per second because otherwise the intersection with the vertical axis would be say at some other value and that would be because after all v naught corresponds to t equals zero so you have some other value of initial velocity and then from there right with you know any numerical value of acceleration it'll always be a line the equation of a line just that steepness of the line is going to change as the acceleration does okay also the line can literally point the opposite way in other words downwards if the acceleration is negative the sign matters okay okay but here's an example how about an acceleration that is greater in magnitude and has a non-zero initial velocity might look something like this okay so this would just be the velocity function for some other velocity okay still growing with time a little bit steeper and a non-zero initial velocity okay but every single time it's the equation the form of the equation of a line and what i mean by that for those of you to kind of remember this from a math class it's the idea that y equals mx plus b right so in or in this order it would be b plus mx okay if you've seen this as an equation line i just want to leverage that familiarity right because what we're seeing is that y is equal to you know this um you know of course the vertical value okay x is the horizontal value y is the dependent variable dependent variable x is the independent variable and b is the intercept okay while m is the slope okay v naught the initial velocity is the intercept acceleration is the slope t is the independent independent variable and instantaneous velocity v is the dependent variable y see it okay so constant acceleration uniform acceleration leads to a case right where we have equation of a line we have a linear velocity okay right oh here's another example of uniform acceleration we'll get back to that one though okay so what happens then to displacement in this case so velocity is linear displacement is parabolic okay plain and simple the distance at any instant in this case right where velocity is growing linearly is going to end up increasing with the square of time okay because you have this case where you know if you think about it there should be a non-linear increase because you're speeding up right so you're covering more ground each increment of time so that means that the rate at which you're covering ground is increasing that's what it means to be speeding up after all so that means that if you're going to track displacement over time which is graph is right displacement or position as a function of time it should get it should be getting steeper you should be covering more and more distance you should be having a greater greater position relative to zero as time progresses progresses okay that's what we see here and it turns out the exact mathematical form is a parabola okay that's why they make you study parabolas so much because it's so important to constant acceleration which is a time of newton's laws hundreds of years of relevance okay one of the most important laws for understanding motion a lot of people care about motion okay not just physicists so the distance traveled is equal to the area under the velocity graph okay we've talked about that in the previous lecture i gave a good explanation including talking about the units or the physical dimensions as they're called of the area under velocity graph velocity versus time and why we know that that is displacement okay so refer to that video if you like but we'll take it at face value here we'll take it at face value here okay so for example if it's you know if it's linear right if your displacement is linear then the area under the curve is just going to be you know the area of a or rather if your velocities is linear then you have a triangular area underneath your velocity versus time function that area is displacement okay so let's think about that then right so if it's a triangular area what does that mean then about how much area is being covered right so if a car starts out with zero initial velocity the final velocity is at the average velocity okay is one half times a t okay so the idea is that because velocity is linear and this was this was highlighted although i didn't pay special attention to it over here but it did in the previous graph okay it was one thing i kind of meant to come back to was it shows here this this v with a bar above it um if you're reading a textbook then you are familiar with this this is average okay in this case average velocity the bar always denotes average so seeing the average here means what we're seeing is that it's one half v well v is just the final value at some corresponding final value of t okay so what's saying is that the average value is exactly one half of that final value how do we know that well because it's linear okay now to get this exact clean result we need to have our initial velocity be zero but it's still a really nice result and it only applies to a this linear relationship linear relationships are so incredibly kind of simple that they're incredibly desirable in terms of actually modeling systems using mathematics to understand the physical world now they're also fairly accurate as as is the case here okay so bringing that up right so we have right that average velocity is one half the final velocity well the final velocity is just whatever velocity you've reached after a certain amount of acceleration times time right because we saw that the area under the curve of a you know um of a velocity versus times function is displacement well it turns out that the area under the curve of an acceleration versus time function is velocity that's all we're saying okay you you're following me that's all we're saying okay so let's go over to the one that had constant acceleration i'm gonna switch back over to here right so if we look at this graph right here we look at constant acceleration all i'm saying is that we have some elapsed time so you know some t you know initial to t you know final and within that time frame from the time that the clock starts to the time the clock you know hit stop that that area underneath the acceleration versus time graph that that area is going to be the change in velocity the magnitude of that area is the change in velocity okay because the height is acceleration okay the width is time when you multiply acceleration times time you get velocity you get it from that linear relationship right because v just equals v naught plus a t if v naught is zero to keep things simple right just you know just set it to zero then v just equals a t right therefore delta v just equals eighty okay see it okay you know so that's that's what we got we got that that area includes that rectangular area so very simple area to to calculate and and visualize is just acceleration times time okay so right that's what we're seeing here is we just replaced v with a t so we have the average velocity is one half a t okay well then we go back to same argument as as then this would be covered in the previous video looking at a velocity versus time graph knowing that the area under velocity graph must be none other than displacement then we can say that d just equals v times t okay now v times t implies that it's a rectangle right because if it's some other area you have to account for it such as you know for a triangle it's one half base times height which is precisely why we have it over here is one half v okay because it's one half the area underneath all right so then how can we get away with here just making it a rectangle well because it's v average and v average is not going to change within the given time frame because velocity is linear okay so that that's it's cool because you literally you're getting something that you could drive with calculus by the way you can completely get to this point that i'm trying to get to with calculus perfectly fine it kind of justifies the most rudimentary use of calculus but we're doing it just with the basic algebra and understanding of graphs and just what it means to have you know certain axes in this case with physical dimensions to them okay with units to them all right so average velocity is a flat line just like just like constant acceleration because remember average velocity doesn't change okay so then it's just a rectangle average velocity times time is the displacement okay and we know that average velocity is just one half a t so we can substitute it in and we get that displacement is one half acceleration times time squared there's the parabola okay that's remarkable this parabolic shape right here one half a t squared okay it's a parabola it has that one half that comes right out of the formula for the area of a triangle okay pretty cool and so that that they'll always show up in our formulas that one half is real right and it's it's not just there for some arbitrary reason it's part of the math part and parcel of it now but the fact that it's proportional to this the time squared tells us that the displacement you know grows more quickly as time passes which makes sense because after all constant acceleration means we're speeding up so of course we're covering more ground you know as time passes because we continue to speed up okay and furthermore we can extend this formula right and we'll do that with a little bit of hand waving which just means that we won't kind of go back to the very basics and justify it but one thing we did to make this this explanation which i i touted as being you know so elegant that we can just use the graphs one thing we did to kind of make it work was we remember we set the initial velocity to zero i did that just a minute ago in the notes right so refer back to that and we also set the initial position to zero now can we generate generalize this and have a non-nonzero initial position and a non-zero initial velocity absolutely okay and it kind of makes sense you can just add the position because after all that's what this is tracking this is your position function d is just wherever you are compared to wherever you started like a race okay now as far as this fact that you can have the velocity start at some value other than velocity we'll actually end up with another term a term that would be familiar to those that remember the expressions for parabolas because we're going to end up or like the quadratic formula we're going to end up with a term that has velocity times time so we're actually going to have our variable often time will be our variable show up as t and t squared in different terms okay now each term all has the same units but you'll see what i mean in just a second okay all right so here's that idea of extending the displacement to include non-zero initial velocities all right so let's let's see how um how would they describe it here so you get this uh to not be cut off quite as much so for a non-zero initial velocity the total distance covered is the area of the triangle plus the area of a rectangle because you know think graphically right i showed i showed an example of this before i said well it's always going to be a diagonal line right velocity will always be a linear function when acceleration is constant but where it crosses the vertical axis just depends on whether or not your initial velocity is zero okay a lot of times we can set it as zero sometimes we can't for whatever reason sometimes we just want you students to get practice with it not being set to zero but whatever the reason here we have an initial velocity not set to zero so the total area under the under the curve here will be the triangular part which we already justified as one half a t squared and down here the rectangular part is just v naught times t which actually you can pretty you know graphically it does show up pretty easily you can see right because it after all the height is just literally v naught units and meters per second the um width here is seconds right just you know after all the horizontal axis is time and so there you end up with units of meters right because meters per second times seconds seconds cancel just got meters left so you know and it's a rectangle right so that actually is exactly how we end up with this other term this v naught t so you can see then that we now have a formula that works fine for finding how far you've gone relative to where you started right that's what this distance formula is giving us for any acceleration okay measuring meters per second squared that's the a and any initial velocity okay now we've got some examples of it so the velocity of a car increases with time as shown what is the average acceleration between zero and four okay so take a look at the graph what's the average acceleration notice it's the velocity versus time function okay think about slope okay well just one meters per second squared because the slope is one you went up by four in four so it's four over four rise over run one meter per second squared okay so the velocity of a car increases the time is showing what is the average acceleration between four and five much steeper section still pretty straightforward you just need to think about the rise of a run what value do you get make sure you can calculate this okay so the difference right 12 minus four so basically you have that rise of eight over a run of one so eight eight meters per second squared pretty comparable to gravitational acceleration okay the velocity of a car increases the time is shown what is the average acceleration between zero and five okay so just like as we as we dealt with averaging other quantities like speed and velocity we don't want to take the average of the averages instead we just want to compare the end point to the to the beginning point and the total elapsed time so it's going to be 12 over 5 okay which as a decimal is 2.5 meters per second squared okay that's the average acceleration over the entire time okay so graphs i've said this a few times but i'm glad they say it too right glad the textbook agrees with me that they contain a huge amount of information right whether that's the velocity graph which is great because you can interpret you know displacement from it as well as acceleration whether you're thinking about the area under the curve versus the slope the tangent line and you know acceleration graphs are no different they can you know contain a ton of information all graphs right you can get so much out of them in terms of thinking about their values of slope the values of them changing from positive to negative the values of them changing in terms of curvature even which is something that we haven't really touched on that you're not so responsible for but is just kind of in the litany of interesting things you can do with graphs okay all right so the last thing i want to do before we wrap up is just to do a numerical example with this constant acceleration now we're going to get some more practice but this is this will be our last thing that we can do in this chapter all right so let's head back over to here all right i want to have fresh page okay so let's consider a car we keep uh having that be kind of our go-to example and so a car has an initial velocity of 10 meters per second okay on the freeway okay then the car because maybe it wants to pass the truck is going to accelerate at 4.8 meters per second squared for four seconds okay all right so the car hits the gas accelerates at a pretty big rate this would be half of gravity so you really feel it you push back into your seat with half your weight that'd be a big acceleration okay so the question then is during that four seconds of of of acceleration essentially what what's going to be the finals the final like conditions right so i have two questions one is going to be what is the final speed of the car okay i don't say after four seconds but that's when the system ends okay so you know it started at zero seconds ends at four right and then the other question is how far does the card travel okay so i'm gonna go straight to the formulas here um you know we've seen we've seen we've certainly seen the graphs ah never mind at least put the grass in a little bit it's like ah am i belaboring the point but now we'll throw a graph in just because it only takes a moment okay so the idea then is we're going to have a constant um acceleration okay so this is going to be our constant value of acceleration this would be acceleration as a function of time meters per second squared over here seconds and our value which doesn't change is just 4.8 okay meanwhile our velocity is going to be linear our velocity doesn't start at zero because it starts at 10 meters per second right and then it's going to increase at a linear rate okay the exact kind of value of the slope i'm just kind of guesstimating here but this would be velocity in meters per second this is time in seconds this is our initial velocity v naught which we know is just equal to 10 meters per second okay we're at that linear function that v naught equals or rather v equals v naught plus a t so it'll be exactly this v equals v naught plus a t okay now we didn't have an initial position nor were you asked to worry about that so your position graph okay which will only be a matter for number b is going to be a parabola okay we only care about you know how far you've gone in the four seconds so we we definitely set our initial um value of position equal to zero right doesn't matter to think of it as anything else right and this is just going to be d in meters this is of course again going to be t in seconds this is our parabolic relationship and our function here right is just going to be that however far you go in that elapsed time is just going to be v naught times t plus one half a t squared okay so the fact that i gave you the time and i gave you the initial velocity and i gave you the acceleration means that this particular version of this uniform acceleration problem is clearly in the plug and chug category which is a great place to start okay so what we do then for part a is we'd say okay well we just want to find the final velocity okay which is the instantaneous velocity after four seconds and it was just it's just going to be v naught plus a t or here it's going to be 10 meters per second plus 4.8 meters per second squared times 4 seconds all right so over here i'm just going to go ahead and plug that in so it's just going to be 10 plus 4.8 times 4 because i can't do that in my head and we get 29.2 meters per second so we more than doubled our speed or velocity if we care about the direction but we more than doubled that value within the four seconds okay so we you know it's a significant increase right because you know the acceleration is you know an extra 4.8 meters per second per second and so we had you know four seconds of that so if you think about it right here is essentially four times four and then some so you know that in other words this product right here is greater than 16. so 10 plus 16 that's right that's how we're getting something greater than you know then 20 essentially but you know we more than doubled okay okay and then finally in part b uh we would just want to find the final position so then d is just going to be v naught times t plus one half a t squared that's just how far we've gone during our four seconds of acceleration so then we'll just put in our 10 meters per second because it matters how fast you're going initially you know and then times four seconds plus one half times four point eight meters per second squared times four seconds and then square it okay so let's go ahead and plug that over here into our calculator so we got 10 10 times four okay and then plus one half all right times our acceleration value of 4.8 and then times 4 squared so 78.41 meters okay you still see that yeah so 78.4 right meters that's how far we go that's how fast we're going all right those are the two questions asked okay so hopefully that really illustrates some of the things you can do here with um with you know a use of this particular equation justified graphically describing a very important situation of constant acceleration which we'll learn comes from a constant force in more detail later and will tie in with the very relevant situation real world situation of free fall in the very next chapter okay so that's it for chapter two i hope this has been interesting and informative thank you so much for watching these videos