hello and welcome to this third video from chapter two from the physics of everyday phenomenon this is griffith book here all right so so far we've covered an introduction a lot of related ideas talked a lot about movement in general velocity and speed specifically defined vector okay but i'm not going to go over that again because we need to move on to our next concept here acceleration which i mentioned many times what is it okay so it's a rate a rate of change we call that a rate right what is it a rate of change in velocity over time okay so it measures the rate at which velocity changes per time okay so that's interesting because we don't perceive velocity because as we'll learn if you're moving at constant velocity then there's no net force acting on you now we're not moving at constant velocity we are orbiting around the sun our planet is spinning around its axis of rotation however those resulting accelerations which are centripetal accelerations more on that later those accelerations are pretty like marginal we're not going to notice them at all in terms of the dominating much larger force of gravity so we don't we don't notice that we aren't moving at constant velocity in other words when you sit in your chair for your abil your ability to perceive you are moving a constant velocity therefore you don't feel the movement you don't feel the orbit around the sun okay because if we were in fact moving in a straight line at the same speed that we're orbiting around the sun then we truly couldn't feel it okay all i'm saying is that there's a small centripetal force because velocity is changing because our heading is changing because we have to complete that circle which is our orbit okay but we definitely feel accelerations that are bigger than those one acceleration that we feel of course is falling the acceleration that's caused by the force of gravity we feel accelerations during quick stopping and speeding up in cars right many direct mini examples of accelerations elevators when they stop and start they kind of have a startling effect right our body responds to that okay so we definitely notice accelerations now what are they they're just a change in speed speed is the magnitude of velocity so that's one way you can get an acceleration slowing down or speeding up or a change in heading in direction of motion so if you were heading directly north and then you angled two degrees to the west of north will that change would be a form of acceleration okay acceleration is a vector just like velocity and of course you can change direction and magnitude of velocity simultaneously in which case you know that that would just be a total acceleration okay oh we're going through these slides sorry that okay all right so there's a famous expression a silly one that says it's not the fall that hurts it's the sudden stop at the end okay now the fall you are right you're accelerating certainly now but you're not going to experience any forces because everything around you is accelerating at the same rate okay if we're thinking maybe the case of the elevator in free fall right that's in a crash into the ground at the bottom of the elevator shaft okay so in that's in that case then when when are you going to experience a force where you're you're actually going to experience force during the much larger acceleration which is the sudden stop at the end the free fall is a certain acceleration one that will learn well which is 9.8 meters per second square that is the acceleration shared by all objects with negligible air resistance near or on the surface of earth 9.8 meters per second squared okay gravitational acceleration of earth now that that value right that's significant amount of acceleration things speed up pretty quickly but when the elevator comes to a sudden stop at the end that acceleration is going to be many times that right maybe 160 g or something and we can't withstand that much of excel that much acceleration so when you think about something hitting the ground if in a less morbid case let's think about like a fruit right you drop like a ripe fruit on the ground and it splatters in the ground you really see that it the the structure of that fruit could not withstand the sudden acceleration that resulted in stopping suddenly upon contact with the ground now obviously there's a lot there to think about what's going on with the forces the force exerted by the ground on um onto the fruit there's also a conservation of momentum because essentially the ground is unmovable so it has no you know significant or measurable amount of movement so we have that although all the change momentum ends up being imparted to the um well to the um fruit right and that that's directly related to forces as well so those are key ideas momentum force but come bringing it back to where we're currently at with acceleration well it is just a sudden acceleration and that's actually a great way of thinking about it because you could have the same effect of the fruit you know liquefying if you just if you reach that same magnitude of acceleration but more gradually it's just not something that you know you could do in everyday situations but imagine you're in a spaceship and opposite of every day right and this is some futuristic spaceship so you know relatively so um no breaking of physics or anything so no warp drives just fast and it has some sort of you know fusion drive or something and it's just accelerating steadily for a long period of time okay and it's accelerating let's say at a rate of 1g well if you're inside that spaceship then you would essentially feel artificial gravity the acceleration of the space spaceship at that 1g rate would give you a concept of weight okay the rate of g one g is a is a measurement of acceleration by the way that's a meters per second per second okay so um here let me show you just that like the units of it right so you can see where's the first time there we have right so that we see the units of acceleration meters per second per second so just meters per second square okay so that that value that we take for granted on earth is 9.8 approximately 10. okay so here's the idea though about this you know it's the sudden stop being that's where we're at that's what i'm elaborating on here is that sudden stop well in this in the case of the spaceship it's not a sudden stop it's it could be gradual so now that spaceship that was accelerating at one g let's say now it it turns it's you know it's drive it's it's you know yeah it's thrusters up in terms of how much fuel they're burning and then they generate a greater force that causes a greater acceleration because you know say it's the same mass of the spaceship right that's newton's second law by the way that relationship between force and mass and acceleration the term we're just just introduced okay now that you could maintain that right but i said i said we're not we're you know we're burning more fuel so now let's say the acceleration goes up the 2g 3g well it turns out that those those humans by the way could survive that you would feel twice as heavy at 2g right which is about 10 meters per second squared you'd feel three times as heavy at 3g which is 30 meters per second square i know i keep saying that it's a real common shorthand to say g right but that's just a measurement of acceleration okay all right free fall acceleration so but you could keep cranking it up right and at some point and this is something that that you know astronauts prepare for uh because when they're when they're going up to the atmosphere they experience values i think around 5g i mean i look it up before this um so i don't know i know it's certainly less than 10 because it you can you can push up to that double digit of g's but because it starts becoming serious seriously fatal or a serious chance of becoming fatal honestly there's a lot that can go wrong um biologically speaking because we're made of you know complicated fluids and you know on a small scale and those are the ones that end up um you know failing at those incredibly incredibly high accelerations in the same way that there's not a you know a health failure in this case in this in this kind of you know upset tummy from the sun sudden acceleration like from a roller coaster right this case it kind of shows from an elevator but imagine this is a more you know more dramatic case right although i'm sure some people get sick on an elevator point being is that that's just you know a disruption of the you know those fluids in your stomach they have what's known as inertia they want to stay in place and when when they accelerate then things get moved around and then your nerves respond to that movement now if you're going to be going up to you like you know as i said up to the human limit of let's say 8g right but you know that's well beyond safety i don't i don't think people that actually get launched in the space experience that i believe is as i said around four or five point being is that as you go up into those those higher values of g it's not about just being unpleasant it's about losing consciousness and then uh one significant thing is having a brain aneurysm because of some failures of key uh blood vessels okay or maybe it's even a building up of bubbles in the blood i'm not sure point being though is that if you do that with the fruit they bring it back to the fruit right so if you and that sudden stop that is you know the the thing we care about so if you bring it back to the fruit then you can have that same very significant acceleration like dropping it on the ground let's say that one you know maybe is 100 g right and you could you could slowly go up to 100 g and the fruit would just get more and more squashed right because it just essentially become heavier and heavier and so you could have the the same effect that we think of as a sudden splattering of the fruit in slow motion right by just cranking up that acceleration right because that's that's what really matters here and it sort of breaks down you know kind of what we think about in terms of you know a funny expression like this you know it's a sudden stop well that really means something that's a way of interpreting the world right and that's what physics is all about okay so acceleration as i said is a vector just like velocity okay um there are lots of examples where you can look at uh like graphs of um we're actually more of a figure right this would be a graph in the sense that it's an xy plot like a coordinate okay this isn't like a graph of versus time which i've started showing you all you're going to see more of in just a minute um so you could you could see that vector expressed in in terms of its xy coordinates the length of the vector is its magnitude that would be its speed if it's a velocity vector um and then when you when you look at that vector that's that tells you it's heading right so maybe you know x represents um west or east and then your y axis might represent north right or you know just whatever whatever you know your graph is i have some examples of that you as you can see wikipedia is in the background and they've got you know lots of examples of vectors beyond the scope of this class but this is a very typical uh you know what how you would draw vectors between um you know just on an x y plot okay so this is just you think this is y as a function of x or really in this case it doesn't matter which is which these are coordinates in space right both y and x would be measured in meters and so what you end up with is you this uh what's delta v delta in all mathematics and physics is going to represent just the change of so we're looking at just a change of v here and well by definition acceleration is that rate of change of velocity so acceleration is delta v over delta t okay so that means it has to point as a vector okay if these vectors point somewhere that's part of how they're defined what they tell us mathematically well the acceleration vector has to point parallel to the delta v vector right in other words the delta t can't affect the direction because time is a scalar okay time doesn't have a direction okay importantly it can't be negative okay so since it can't be negative it can't or you know have some uh it's own heading okay certainly not an x and y it's not spatially dependent time isn't right well then you you know you end up with a situation where you have the acceleration vector being parallel to this one right here so in other words the acceleration vector would be some other length right because its magnitude change right depending on how big delta t is either greater than one or less than one right well then um that would either make it longer or shorter than delta v the point being though is it would point in the same direction right very importantly all right so that's that's you know certainly what what we talk about when we talk about the acceleration being a vector um right here other other examples of it actually showing that acceleration vector in terms of these things called components because you can turn any vector into right into a right triangle this is crucial for really understanding them and doing mathematics with vectors but we're not going to do that because it would require a lot of use of sine and cosines and tangents um and just um not math that we need that we have time to spend on in this class which we that we need to right because we can appreciate physics without it okay the point being is that is common common vector idea um this this uh nice little animation shows you an acceleration vector of a very typical thing this is like this is a pendulum you could think of this as a grandfather clock swinging back and forth and what we're seeing is we're seeing instantaneous vectors right so you know simulated with a computer and instantaneous is always relative it's like how many times per second is are we making the calculation enough that we assume it's approximately instantaneous so we call it instantaneous it's not an average over you know a significant amount of time so and that so anyway those are the vectors that we're seeing we're seeing an instantaneous factor in blue of la of velocity okay so the velocity is like the heading of the ball right so when the ball is swinging up the velocity vector points up right then notice the velocity vector momentarily disappears at the top of the swing that's because this pendulum right at the top of the swing like a child on a swing is momentarily at rest at the high points right both in the front and the back okay um the velocity is largest at the bottom of the swing which also makes sense right that's we've picked up the most speed uh as we'll learn when we talk about energy that's because you converted potential energy gravitational potential energy into kinetic energy at the bottom of the swing so we see that here velocity vectors biggest okay but acceleration is what we're currently talking about and we can see the acceleration has a pretty a much more complicated effect that i can't um spend as much time on or we already go off on enough tangents right i do point being though is that it's changing okay the acceleration vector has a effect here that is that is dependent not just on the length of v but also where v points so as v changes in terms of you know kind of growing quickly in magnitude versus changing dramatically an angle that causes the acceleration vector to do behave differently um in other words the acceleration vector at the bottom of the swing is behaving almost entirely in fact entirely to approximation as a centripetal acceleration and at other points in the swing in particular at the top and the bottom at the high points well um so really not the top and bottom the left and the right the high points say that correctly at those points the acceleration is behaving entirely as what's called a translational or tangential acceleration where it doesn't depend on the rotational motion at all it's it's not a centripetal center seeking acceleration it's just driving it you know str in a straight path but that doesn't last because this they just think pendulums have a whole interesting mathematics about how they how they function um as a little wedge of a circle okay which ties directly into the angles of the triangles of the vectors themselves so it's pretty pretty uh amazing system now the last um thing i wanted to show you here um was the so definition of acceleration and just um oh and just be a little off the animation that shows at the top you know fundamentally you know what one thing that when we think of acceleration dropping something right we release release this yellow ball here and then the little animation and well at first it you know starts from rest and then it speeds up as it falls okay that's that's the acceleration you know at sets is just kind of fundamental to what we understand and that that's what we're seeking to explain um with the language of mathematics and the rigor of physics okay all right so okay it is right vector points in the same direction as delta v we saw that example of that all right it refers to any change in velocity i've said that many times doesn't have to be a change in speed it doesn't have to be a change in direction it could be one or the other or both okay in phases we even refer to a decrease in velocity slowing down as an acceleration we would call that a negative acceleration in a 1d case or in a 2d case maybe one component of it is negative but it we do not call it a deceleration right that term it you know we might show up might show up in a word problem although i might remind you if it does that that is a negative acceleration right this is pointing in the negative direction so the direction of the acceleration vector okay now i think one-dimensional like in the figure down down here right everything is showing up in a straight line all right is that of a change in velocity if the velocity is increasing the acceleration is in the same direction okay so if we look down here we're seeing that the acceleration vector okay is pointing in the same direction as these velocity vectors right vectors are arrows okay vectors have some values some units right both these would be meters per second the velocity vectors would be right but their length and direction matters here we just care that they're pointing forward and we do care how long they are okay so then this is the initial velocity this is how much the velocity changed this is the final velocity the final velocity is just the sum of the length of these two vectors these two arrows right sum them together you get that longer one okay the acceleration is not directly related to any of these lengths it because it depends on how much time passed because remember the acceleration is not just delta v right in terms of its value it's delta v over delta t so if you actually have a numerical value you have to know that delta t and we're not we don't we're simply not given it right we do not have that information here all we can say for sure is that the acceleration is pointing in the direction shot okay now what if the situation was different what if instead the car started at 18 and so it had a suitably larger initial velocity so something that looked like this okay that's v initial and then at the end it's then going to have some speed of we'll say four only right so went from 18 to four so it's slowed down and then in this case the u would just be some little small final velocity like this okay so how would the acceleration vector look in that case well if you think about it your v initial is large and now my alternate example right this is our new v initial our v final is quite small so the only way to sum up you know a third vector so you know sum up something with the this v initial in order to get that suitably small v final we'd have to sum up something that points yup you guessed it backwards so our delta v is negative okay so in this case where we go from 18 to four where the the velocity decreases our delta v is less than zero we have that negative delta v okay would that make a negative acceleration you betcha heck yeah because the the delta t can't be negative right change in time is always positive so anytime your change in velocity is negative slowing down that's a negative acceleration okay and remember try to not use the term deceleration okay so the direction of the acceleration vector is that of the change in velocity all right if the velocity is decreasing the acceleration is in the opposite direction all right there we go right and did i repeat myself heck yeah because it's the same statement you know it applies it doesn't it doesn't matter whether it the acceleration is opposing or parallel to the well the initial velocity vector okay now the other case you know where things get two-dimensional but is kind of a good simple place to start is thinking about circular motion right so think about a a car in this case bird's eye view going around a turn all right so in this case we're not going to overly complicate it so we'll keep the length of the vectors the same so v final and v initial have the same length right so the speed is constant that's saying the length of the vectors is unchanging the magnitude of velocity is constant the speed is constant those are all equivalent statements all right what is changing is the direction okay so what we end up with is we do we do this idea we do this process where we think about vectors and adding them up tip to tail and in this case we're subtracting one because what is delta v right delta v by definition is right over here so delta v is just v final minus v initial okay as numbers that's fine and they would work as numbers great if we were in the 1d case like we were just seeing before because in that case we could we could just treat them like values on a number line and we actually don't need to really see that we don't need to visualize it other than it being helpful which is why these slides are included okay but when we're dealing with this two-dimensional case you can't avoid it so we when we add up the vectors and we want to not not have a value here because we didn't we we would have to at least use the pythagorean theorem but just to have a an idea of where it points we can just do this idea of tip to tail vector addition okay so it's tip to tail vector addition and all that is is what's shown here okay so if i was to add the two vectors okay v initial plus v final so v initial was pointing this way i'm just trying to draw it similarly and then v final we can see kind of kind of points down all right so this would be the way you would add them together this is the initial okay this is v final and why does that follow this this rule this term tip to tail because the tail of the second vector touches the tip of the first all right so that's the tip to tail okay tip tail so that's how you add them together not tail to tail not tip to tip tip to tail and that works for any vector all the time okay that is how you can graphically add vectors and you could you know if you did it accurately right and then you use like a protractor and a ruler you can actually get measurements right before calculators are readily available people did this by hand and got pretty good measurements right for hundreds of years okay but we don't need to do that right but it's still kind of it's good to think about that i think it helps understand this mathematical idea which is important to physics right vectors are going to keep coming out all right so but we're not adding them together because delta v by definition is subtraction right well when you subtract a vector that's the same as adding the you know the vector that points in the exact opposite direction so 180 degrees the other way all right so adding you know negative v initial is the same as just having the initial go the other way all right so what what we'd end up with is we would simply have the v initial vector pointing the opposite way all right so this one instead would point this way all right and then we would add them up like that now when you do that that is the same in practice as putting them tail to tail all right and so subtraction ends up being the same as sort of tail to tail alignment and why is that well look right if i if i took the initial and i put it the opposite way so in black well it'd be more clear to choose a different color let's make it orange right so right on top same length right same direction the only difference is the well you know the same direction as in the same angle but more importantly is it's exactly 180 degrees the other way because what is this orange vector this is negative v initial okay as you can see that's the one that needs to be negative okay according to the definition of delta v change in velocity okay change velocity points in the same way as acceleration that's why we're talking about it right so that negative v initial right will look now if i'm doing tip to tail addition i'm going to take negative v initial plus positive v final because it doesn't matter you know what order it's in right taking negative v initial you know plus v final is the same as this right see same thing so you know that said right what's going on well then i end up then with tip to tail addition just getting delta v right so what does it tell us right well it tells us where delta v points where there's a point keeping these you know the the angles exactly um or keeping the lengths of the vectors unchanged because we didn't change speed where's the point exactly to the middle of the circle okay so the delta v by following tip to tail vector addition points right to the middle of the circle therefore so so does the acceleration what have we shown with just this concept of vectors and definition of acceleration why the centripetal acceleration is a thing which is pretty neat right that's like a big sort of idea right we show that there is this fundamentally there's an acceleration that comes straight from the definition from the map that points right to the center of the circle for anything that's following a circular path okay or approximately a circular path and what's the name of that acceleration that points right to the circle the centripetal acceleration why does that matter because that's going to explain things like you know natural orbits of you know like moons and planets as they explain things like you know man-made orbits of satellites um and you know the international space station that's going to explain things like understanding um you know where you know how how forces are balanced in terms of you know like a car going around a turn right not slipping off of the turn and so on right so so it's such a big idea with so many implications so many forces can create that centripetal acceleration right centrifuges that have medical applications right in terms of separating different different materials right they have different densities okay that's all going to be driven by a centripetal acceleration and the resulting centripetal force okay or maybe more from the perspective of the specimen the centrifugal force there's a difference okay all right but moving on terms that we like to define that we do the same thing with speed and velocity is we have average acceleration okay average acceleration is exactly that it's average over some large amount of time okay so some significant amount of time you'll be given given the delta t it's the difference from an instantaneous value okay it's the secant line instead of the tangent line okay we talked we talked about this before right we talked about this idea right secant is the average see secant this case it was the average velocity because notice the vertical axis is distance right but if i just you know replace this vertical axis with velocity velocity well right away what's the what's the curve right what's the the orange curve well the orange curve now isn't a velocity curve it's an acceleration curve because velocity is a function of time the rate of change of velocity versus time is acceleration this graph is acceleration now the secant line is what is it average acceleration because it's averaged over some large gap in time some delta t okay but if you make delta t you know so small that it's unmeasurable then you end up with something called the tangent line or approximation of it okay and the tangent line right what's that that's the instantaneous acceleration okay all right so going back to average right so if you had for example a change in velocity of 20 meters per second um you know maybe that that change you know was noticeably varying over those over some of the time which here is given as five seconds but you don't care right you just want to average it right so all you need to know is that you have an overall gross change of 20 meters per second maybe you started a zero you ended up 20 right and it took five seconds to do it in that case you just you get your average acceleration your rate of change of velocity per time meters per second per second okay so that's the same as just saying meters per second right times one over seconds because you divide you know you divide by something a second time that's same as you're multiplying by its reciprocal so what do you get then right you end up with meters per second squared which is the way we're always going to see the units of acceleration written okay and even if you're going to say it as a sentence it would be distance per time squared okay it's acceleration that's what it is it's a rate of change of velocity all right so right so this is this average acceleration we don't care what happens in the middle we're not you know we're finding our measurements down to the instantaneous values it's just a you know an approximation a gross approximation in the sense right because we're just looking at some in this case a five second gap right a lot can happen in five seconds but that's what averages are all about okay and hopefully that also kind of lays down the basics of what acceleration is all right so instantaneous acceleration that's the tangent line okay all right and it's that it's a acceleration at some precise instant right we assume to be a precise incident but we approximate to be a precise instant okay so it is the rate at which velocity is changing at that instant okay because it's the slope right the slope is the rate of change of velocity per time meters per second squared units of acceleration all right so it's found by calculating the average speed over a short enough time right that it's approximately this instantaneous okay so let's consider practically graphing this all right so um there's some slides that are going to kind of be referring back to velocity i'm going to go through those quickly so we can get to the final idea with looking at a graphical represent representation of acceleration and then i will be wrapping this video up because the last one the fourth video in this chapter will will then approach or address a very common case of constant acceleration uniform acceleration and just uh pay special attention to it and the equations right so you're really gonna have your first set of working equations okay but first let's get to these graphs okay so um we're gonna describe a car's motion imagine we kind of tracked it right these are data points that we tracked along um you know with some some technique right maybe we recorded the video and tracked it um based on the known frame rate of the video so we we know at some time the exact position of the car right and then we know at some other times right not huge you know precision here we're just measuring you know down to one second but we got some information okay so what does that give it give us well it tells us the distance in centimeters right this car is moving across the table top or something on the floor and a minute a total minute of movement car's not moving that fast right so when we look at this graph right since this is distance versus time the slope here is velocity so you can see that in this this case these aren't like nice natural curves these are just kind of chunky data that was collected in a rudimentary way well so when you so that means that over this whole segment from about you know from actually from 0 to 20 seconds the velocity is approximately not changing which is why it's just a straight line okay so this would just be a big segment of constant velocity constant non-zero velocity this is the car moving forward then okay the car is going to get to a point where the distance you know sits at 30 for uh was it 10 seconds yeah for 10 seconds what does that represent that's when the car stopped right because if the distance is remaining unchanged over some period of time that means the car is not moving right so that means that here the tangent line okay which should also be the secant because there's no there's no real kind of you know because it's it's straight from multiple multiple ways right but here the tangent line is just flat that means that the slope is flat which means zero velocity okay so this would be a zero velocity for that segment of time from 20 to 30 seconds okay so then right then we have another slope right we have another region if we were to average over you know say from like you know zero all the way over to 50 okay there we would actually have to draw a secant line because we'd have to take the overall average okay we wouldn't want to say averaged our three you know periods of constant velocity because they're not equal averages we'd have to do a weighted average so we instead just want to do this okay then that's a throwback to when we define how to actually calculate average velocity okay and the common pitfalls of doing it um so refer back to video two from this chapter all right so right when is the car not moving we spoke about that it's going to happen in this period of 20 to 30 seconds at what time does the car start moving in the opposite direction do you see a point where the car is moving in the opposite direction yeah absolutely right negative slope right this is velocity less than zero that's a car has here came to rest right so here the car came to us and stayed at rest for 10 seconds here the car just came to rest for like one quick second it's like you know like the car like you know was like slammed on his brakes came to a stop but then immediately like kind of started backing up so there's just like this one like one quick moment of rest and back the other way right so one instant of rest and then negative negative slope means negative velocity which means moving backwards right okay all right so the slope um is the change in the vertical quantity okay so here as since this has been sort of graphs that have honestly been reviewed for us right um in reference to acceleration at least so you know here since the vertical axis has been distanced all of our slopes have been distance per time so all of our slopes have been velocity as i've been saying okay rise of a run okay so at what time does the car start moving in the opposite direction what precise time remember it's that instant where it's at rest 50 okay right at 50 seconds okay then it continues to move in the opposite direction from 50 to 60. and then our data stops or maybe the car just completely comes to rest runs out of batteries okay so we could actually then look at these segments of velocity over this period and we have these chunks of velocity either because the car was literally moving in sort of this jolting way right which is quite possible for a toy car or it's because of kind of the the way that we collected the data we weren't capturing everything right but for whatever reason we have you know this big segment of constant velocity right so if we were actually to graph velocity that big segment would kind of almost look like a bar graph but it wouldn't be it this would still be caught you know instantaneous velocity or some rough approximation of that right but we just don't we just know that it remains exactly constant right at a value interestingly of one centimeter per second and where where we get that one centimeter from right we actually get it from from doing a calculation right so if we look here and we say okay well we know that this right here is 20 seconds and we know it's going to correspond to 30 centimeters right so it looks like almost a little bit above 30 centimeters uh what do you think more like a not a 35 but maybe a 33 right so i'm sorry that's a 15 centimeter so 16 we will say all right so we'll say that this is 16 centimeters so our then if we want our average velocity right so our v average is then just going to be our 16 centimeters divided by 20 seconds okay so 16 over 20 um we'll think you know uh that's going to be four-fifths right so it's going to be four fifths of a centimeter per second right or point eight okay so if you look then well our units of velocity in the graph or example graph here are precisely in centimeters per second four fifths right point eight right as a decimal 0.8 centimeters per second we can see that is indeed what the graph approximates right so i guess my values are pretty accurate and there you go you have that constant velocity for a solid 20 seconds right staying at that value and then suddenly it just disappears right so you have this instantaneous change like it just as you know as far as we can tell it just drops down to zero then the car is motionless for 10 seconds and so on okay so that's what we're seeing we're seeing this jolting motion of the car over these different periods of time we're actually seeing velocity as a function of time okay that's relevant because that's going to allow us them to talk about acceleration so let's you know let's look at finally one more graph that shows distance right displacement over time so um here we're looking position on a car with respect to time does the car ever go backwards okay assume no no u-turn here different graph use the same logic as before make sure you answer it you ready okay right absolutely just like the graph before there is a segment right right at the end where the car is definitely going backwards it's going in the reverse direction here negative slope means it's going backwards okay so is the instantaneous velocity at point a greater or less than a point b right think about that think about the instantaneous velocity remember this is a graph of position versus time the slope is instantaneous velocity okay so it's less right so the instantaneous velocity can be compared right so the steeper slope indicates the greater value right so the value is greater at a okay oh that's what was asking right so it is instantaneous velocity at point a greater yes okay because the steeper slope is this one here see the slope that passes through a this slope right we call that slope a that slope is definitely steeper than slope b right which means whatever those values are depending on the scale right maybe this is you know 10 centimeters or something whatever it may be that's gonna actually give us the values rise of a run okay and we would indeed find that slope a has a larger value of velocity right so this is velocity a velocity b right they're precisely equal to the slopes okay so now let's actually look at another graph that is velocity versus time okay we saw the one that was you know the big um you know look like the bar graph big chunks of velocity this one's a little bit more natural right so we have velocity staying fixed not zero but fixed okay then increasing linearly right and then decreasing linearly interesting so what's going on right so in this graph is the velocity constant for any time interval what do you think is the velocity constant for any time interval okay absolutely velocity is constant between zero and two it's not zero okay that's how we're asking what's asking if it's constant right it's some non-zero value maybe this is one meter per second okay right maybe it's one kilometer per second who knows right but whatever it is there's some value it remains constant for a while okay all right so same graph right during which timer interval is the acceleration greatest now hint here if you want a hint if you don't then you know don't look at this part but the acceleration remember by definition is delta v over delta t okay at least the average acceleration is which is all we need here okay well delta v over delta t if you look at the graph right that is rise over run okay so that means that on a graph like this any graph that shows velocity as a function of time velocity versus time on any graph like that acceleration is a slope plain and simple okay and that be careful the units right because of the velocities in centimeters per second then your acceleration will also be in centimeters per second squared you know so there's little pitfalls there it's best if everything just stays in meters per second because then your acceleration will be in meters per second squared okay and time especially right because if you're measuring your velocity say versus time measured in milliseconds right which is a thousandth of a second be careful right because then you got to think about what what units would your acceleration be will be milliseconds squared all right so that may come up things to think about point being though is the acceleration is simply a slope so i bet now you can answer the question in what in which interval is the acceleration the greatest okay with the magnitude by the way right so it certainly is b right segment b which is between two and four right because why because that's just the steepest okay steepest slope greatest acceleration for this velocity versus type graph okay now a different velocity versus time graph this represents a car that's moving in a straight straight road so one dimensional we say right and we're seeing its velocity versus time does this car ever go backwards okay think about velocity here does the car ever go backwards okay do we already ask this one no this is a different question similar to the one we saw before different okay no it actually doesn't and if you said well hey wait wait but there's a negative slope well that just means that it has a negative acceleration so this would be an acceleration less than zero what does that mean that means the car's slowing down so on this graph remember the vertical axis is velocity the only way that this graph could have shown backwards motion is if our graph right you know if our line here if it maybe you know we did something like this then you know remain constant velocity for a while and then almost column was constant crossed the horizontal axis because now in this segment here this negative velocity totally allowed it's not negative time totally makes physical sense that actually is the car going backwards okay negative acceleration is not plain is simple right now negative acceleration on a car that has negative velocity means that the car is going backwards and speeding up in its backwards motion okay but the negative acceleration didn't tell us that it was going backwards the negative velocity did which is really interesting the acceleration actually only tells us if an object is slowing down or speeding up if we know the sign of the velocity because as i said a car with negative velocity so like in a region like this if it has negative acceleration as well then it would actually be curving rather than a straight line here below the horizontal axis it'd be curving and getting steeper let's say something like that see how it's curving getting steeper downwards because that means that it also is experiencing negative acceleration which is causing it to get greater and greater speeds but all still backwards so all still negative so that's interesting that's a negative acceleration that's actually causing the car to speed up why because the signs of the velocity negative see negative and the proposed acceleration also negative since they agree that causes the car to speed up okay whereas if we have a car that is moving forward by definition positive velocity right above the horizontal axis well in that case if the car was experiencing a positive acceleration then we wouldn't have a line we'd have a curve and we'll talk more about this okay and we saw we'd have a curve of the um or if the acceleration itself was growing right so we'll have a velocity right that maybe is doing something like this right what does that mean that means that our accel our acceleration is growing in the positive direction as is the velocity okay since they agree we have the car speeding up rather than slowing down okay all right so another concept question talked about that that example for a while so we'll continue on right so which point is the magnitude of the acceleration the greatest okay same sort of question but that would just with this graph so if you're able to answer the last one you should be able to answer this one this would be another good chance to make sure you understand okay we're just looking for the magnitude of the acceleration for this velocity versus time graph okay so it's absolutely point a because that's where the slope is steepest okay all right during which time interval is the distance traveled by the car the greatest okay so with the distance traveled by the car how do we get that right so you know what are we looking at here right so we're looking at velocity over time right so think about distance right well velocity by definition okay let me show you neat relationship here one of these kind of key relationships how you can take a graph and turn it into a formula especially when there's simple graphs like this that are lines either diagonal lines or even better flat lines horizontal lines okay but here's the relationship so velocity okay is by definition okay distance over time okay and we're asking for a distance okay that's why that's why i bring it up okay well if you look here then you see that the distance is velocity times time well that's interesting because think about the graph right the graph is velocity versus time that means if i was to say draw a rectangle right here right make that kind of the right right shape and say i shaded that rectangle in well that that area encompassed by the rectangle that would have a height with units of velocity right so you know it's weird because we're thinking about a space like a rectangle right but instead of measuring it you know in heights in inches and you know width in inches we're mentioning it's height and velocity so an abstract idea but it works and we're going to measure its width and time okay so what then would be the units of that area enclosed by this rectangle well you can see right velocity times time it's distance okay the area is distance the area enclosed is distance okay plain and simple we can see it works out in terms of the dimensions which are the units we call them physical dimensions because right distance meters okay well velocity was meters per second we multiply that by seconds we see what's happened is the seconds that have cancelled so of course we're just left with meters and meters okay so this is a true equation in physics all equations must have equal units on both sides otherwise it's either just a lie or a mistake okay one of the two okay but right we worked out this relationship we have then that the units enclosed by you know this graph must be distance covered so it's interesting right because i you know i talked about you know i'd say the simplest case which is the rectangle but you can then say that then all these different segments then going to correspond to a few different you know covered areas right so we'd have you know this first segment right that would be the the area of the rectangle i'm sorry the triangle get my uh you know geometry terms correct here so the triangle is not just base times height but it's in fact one half base times height so very simple formula at least right um it doesn't matter there's a right triangle it just works for any triangle as long as you know the base and height right and so the next one would be the rectangle we talked about that would be just base times height this one um it's kind of an interesting shape right the way you'd want to do this you want to split it up so do a triangle that which would just be this area here okay and then plus another triangle and this would work if you actually knew the values oh here's choose another color right but the point being is that if you then sum those up the sum of those two colors then would give you the area in that segment but here we're just asked to compare them that we are actually given values so which is the biggest which is the biggest enclosed area right well i bet you know between two and four okay that is the greatest distance traveled so wow this is remarkable right because when you're looking at this particular type of kinematics graph right this particular graph describing motion velocity versus time yep this one oh here it is velocity first time okay well it's kind of like the the most useful of all the you know because it compared to you know maybe graphing acceleration versus time or graphing position versus time because velocity is in the middle right velocity is related to distance because if you know you know the the velocity over some period of time where it you know conveniently remain constant and you know how much time passed then i just showed that then you can actually you know find the area enclosed by the graph right just based on a simple formula right a simple relationship that distance is velocity times time so you cannot you in other words you can kind of work backwards from velocity to find distance covered as long as you know time elapsed but then you also can work forwards you can find out instantaneous acceleration or average acceleration by driving drawing secant or tangent lines okay and in this kind of chunky example it's kind of the same one or the other right so over whole segments you can just draw a tangent line which is the secant right it doesn't change and what point being is that you've got that that slope right there is your acceleration so you've got area being distance and you've got slope being acceleration all from the same graph okay that's how graphs showing motion tell us so much in physics okay so kind of to wrap this up we're getting right near the end here which is good been going for a while is that you know this is the first example we saw this was the toy car right before we saw other examples and i said this this was very much review because we already talked about graphs of distance versus time a little bit in the second lecture uh from the series right well well let's let's then finally take it back to acceleration right so we thought you know you saw this chunky motion of you know displacement right this resulted in you know big chunk constant velocity velocity dropping down to zero for a while jumping back up to some positive value and then eventually jumping you know right down to a negative value all right because that's when the toy car actually starts reversing which is in this segment here okay because this is the slope being less than zero the slope here being velocity because the vertical axis is distance all right but then can we actually then finally show an acceleration graph we sure can okay an acceleration graph would just be a number of spikes right and depend and you kind of almost want to draw them like you know infinitely narrow but here we kind of are approximating like just a little spike right um this shows up in a lot of actual real data because what does the spike actually show us right the spike shows us the sudden changes in velocity because velocity here right is is the remaining constant at about you know as we saw uh 0.8 right and then drops down to zero well going from point eight to zero that's a negative acceleration because you're going from high positive velocity relatively high point eight right down to zero you dropped right so the you'd have a spike of negative acceleration so you have a spike of acceleration less than zero okay because anything below the horizontal axis is negative know above is positive all right and then you have a spike of positive acceleration where it jumps from zero up to um up to about half a meter per second and so on okay so that's what the acceleration will look like maybe it's not the most beautiful acceleration it's not one of those like elegant cases like the pendulum or or even free fall which is just a flat line but it's kind of realistically what happened for this jolting type of motion and shows you what it looks like okay um here is a more complicated neat example of velocity i was actually speed because we're interested in the magnitude right so it's just speed this is nothing and and this is a car that we imagine this car gets on the highway right um you know it has to um you know turn on the highway so they're going slow on the on that when they're on the on lane right or the emerging lane but then they pick up some kind of you know steady steady highway speed right this is uh you might think they're going really fast but it's kilometers per hour not miles per hour remember right so going pretty fast and they get stuck behind a truck but then they pass the truck so they got to get them going really fast right and then they but then they're able to you know kind of hit another a good cruising speed but it doesn't last long because then since the highway not a freeway big light in the distance tells them to stop so they have to come all the way down to zero right wait for that light to change okay so that's that's their motion one dimensional motion right as far as we're concerned you might be like oh the highway is like waving around but we've like followed the highway made that our line right that's our axis um and so what's happening here is since this is graphing speed okay um you know versus time at any point the tangent line is acceleration so you have lots of cases right so when when the car was cruising right so you can have this kind of big segment where it's maybe slowing down just slightly it's pretty flat you got another segment here right let's actually overlay some of these all right so you've got all right so this you have a big region here where the acceleration is about zero right it's going to correspond to this segment here because the tangent lines is flat flattened out right the passing the passing segment you know certainly outside the car is you know really you know putting the pedal in the middle really speeding up the pass right it really has to push the car hard right this is a large acceleration right so we'll say acceleration you know larger than zero and uh maximum right this is the biggest case right the car is really picking up speed right and then it kind of you know just it once it passes and it as soon as the person you know lets their foot off of the accelerator off of the gas right they're gonna you know coast for just like just a second and then immediately start to slow down right so this this region here of negative acceleration right this one right this acceleration less than zero that that would just kind of happen naturally because you don't want to maintain that same same high speed that you use for passing the truck because you don't want to use up all the fuel you don't want to get arrested or you at least get a ticket and also because as soon as you take your foot off the gas having accomplished the pass you're going to hit get hit by a significant amount of air resistance which you were working against the whole time right because the car isn't just you know accelerating for free it has to be constantly working against friction especially air resistance um and that's you know that kind of built in built into this graph but it's an interesting story to tell right so these are lots of examples of different instantaneous accelerations over this kind of complicated motion shown here as speed okay and if you wanted to find the total distance that the car had covered um from over this period of about 30 seconds well you guessed it it would be the area under the curve right this case would be you know you have to approximate it or something it something but if you were able to somehow accurately measure this entire enclosed area this pink area in units of you know once you worked it out you know kilometers per hour multiplied by minutes right so you want to you know somehow get that over to meters or something or at least kilometers okay probably convert the minutes or over to hours or the hours or minutes and then just end up with units of kilometers so you actually that purple region would represent in units of kilometers again once the hours are converted into minutes well how far the car went or that this whole process kind of right okay so lots of lots of different values of tangent lines one total distance um you can of course you of course find different segments of distance by segmenting it up starting and stopping at whatever time you want right other than the example i did go all zero all the way to the end okay i think this one is our last all right oh this is a acceleration change for a sprinter just another example i'll quickly go over it so a one a runner wants to reach top speed so they're gonna do most of their acceleration at first so looking here this is this is a speed by the way s for speed so their speed uh you know accelerates really fast and then quickly kind of stops accelerating so there's this region of quick deceleration right um or leveling off of acceleration and then the speed just remains constant okay so that whole region of constant speed well any flat region of speed is zero acceleration so that's just the runner maintaining that point that that final steady velocity all right so the velocity graph on the object is shown is the acceleration of the object constant think about this this is velocity shown for some object like the sprinter okay is the acceleration constant yes or no or can we not know is definitely not constant how do we know because the tangent line isn't constant right imagine at first the tangent line is you know it's non-zero right right then it gets less steep right so it started off steep in other words okay then right by the end the tangent line is basically flat so this is an example of a velocity okay that that approached some final state which means that the acceleration because this would be like a1 this would be a2 and this would be a3 right well a3 is the final one so we have the acceleration approaching what zero right the acceleration is going down the acceleration is approaching zero if we were to graph the acceleration kind of simultaneously right on top of this then the acceleration would start at some initial value which is just a1 and then it would just drop down to zero with its own curve okay like that that is definitely not constant okay what would constant acceleration look like or what type of velocity would it result in well that's what we're going to talk about uniform acceleration constant acceleration and i'll tell you it's integrate lines linear diagonal lines not flat lines but it's going to create linear velocity graphs okay which is elegant and they give us some equations we can work with and that will be the conclusion of chapter 2 this last kind of workhorse topic of uniform acceleration all right well i hope i see you all there and i hope this lecture has been interesting and informative thanks for tuning in