kinematics in one dimension going to be the topic of this lesson we're really going to do some hardcore problem solving in the area of kinematics specifically in one dimension and this is where some students begin to have a little bit of trouble but my goal for this lesson is that you really get a good intuitive feel for much of the material as well as a systematic approach for problem solving my name is Chad and welcome to Chad's prep where my goal is to take the stress out of learning science now if you're new to the channel we've got comprehensive playlists for General chemistry organic chemistry General Physics and high school chemistry and on chatsprep.com you'll find premium Master courses for the same that include study guides and a ton of practice you'll also find comprehensive prep courses for the DAT the MCAT and the oat now in the last lesson on displacement velocity and acceleration I hinted toward the fact that we might do some calculations involving two different scenarios and the first is constant velocity where you have no acceleration and the second was going to be when you have uniform acceleration or constant acceleration and then I said but you're not going to be doing any problems where you have varying acceleration well there's one caveat to that if you're taking a calculus based physics class you might you still might not but you might but for an algebra based physical class like the one we're gonna you know the one we're going through here you're not going to be dealing with a varying acceleration so in the two situations we will deal with so again no acceleration again you're moving at constant velocity or with a uniform acceleration a constant acceleration these are the two situations and it turns out there's only a handful of equations for which you're going to do kinematics calculations and what's nice is when you don't have an acceleration there's one equation that's all you got so we'll see that this equation is related to the definition of velocity so but even with uniform acceleration constant acceleration you've only got a handful of equations but it's enough to where some students get to the point where they just are given a bunch of quantities in a problem and they just don't even know where to start well the good news again is that there's a limited number of equations and even if you just kind of systematically work your way through all the equations because you recognize it was uniform acceleration you'd eventually figure some things out but hopefully we kind of demystify this and we're going to give you kind of a systematic foundation for kind of approaching this as well as a little bit more of an intuitive understanding of what displacement velocity and acceleration are and their relationships to each other all right so before we dive into some calculations and some problems that we're going to solve throughout this lesson I just want to kind of look at these lovely quantities and kind of see where most of them come from so if we take a look at the first one here so Delta X displacement equals velocity times time and again this really is just coming from uh the definition of velocity if you recall velocity was equal to Delta X over delta T displacement over time and if you rearrange this you can get Delta x equals V times delta T so in fact some people actually write this and I see some people some textbooks write this as Delta x equals V delta T so if I'm going to leave it as T it's pretty common to see it this way and the idea is that if delta T is T final minus t initial where T initial is zero then all you've got is T final and if that's the if that's the only time point you have then we can just call it t we don't have to actually signify it something else and so that's where this kind of reduces down to this form right here so uh when you've got no acceleration you're at constant velocity this is the only equation you got nothing else to worry about all right so if we take a look at uniform acceleration so now your speed is not constant it is changing but it's changing at the same rate the entire time and you've got a very similar looking equation where you can't just use the velocity because it's changing so it's it's you know it's either always going up or always going down it's not at one set point but what you can do is you can use a directly analogous equation in which you use the average velocity and if you're calling the last lesson we actually came up with an equation for the average velocity where we say that the average velocity was equal to V initial plus v final over two or one-half times the sum of V initial plus v final or something like that also one thing to note I said you might see V initial and you might see V naught and all my acquired equations are written with a not so I'm going to try and use that here but again that essentially means velocity at time zero so when you see that little not symbol it means at Time Zero all right so you know if we can calculate the average velocity because we know both the initial and the final velocities then we could use this equation and we vary analogous to what we're doing with no acceleration now this next one's a little bit different here and so it turns out it's not going to be easy for me to show you where it's derived from at least not the second part of it because it actually does come from calculus it involves taking a second derivative which we're just not going to do in this algebra based class but we could so the idea is this so you know if you ignore that last term this looks pretty much a lot like both of these so but the key is that you know if you just use your initial velocity well you know the idea is that that velocity is changing the whole time it's either going up or going down and how do you account for that well that's what this last term does it accounts for it and so if you're speeding up so then along with that initial velocity times time you have to add an additional term to get an overall higher displacement because your velocity been going up so you're going to go further and further and further and definitely further than if you just maintained that initial velocity the entire time on the other hand if you're slowing down well if you're slowing down that would actually mean you have a negative acceleration we'll get into this a little bit so but if you're slowing down that means your acceleration and your velocity point in opposite directions and that needs you may need to make one positive and one negative and it's customary at this point to make the velocity positive and the acceleration negative and so in this case you'd be plugging in a negative number for a right there and so you'd end up with this first term and then you'd be subtracting or adding a negative term overall now one thing I just want to bring this up and give you one caveat is that a lot of textbooks will take this and give you a fifth equation and they'll have V not T minus one half a t squared and they'll say well always plug in a positive value for a you just have to choose the right equation if you're speeding up it's this equation if you're slowing down this equation if you're accelerating at this equation if you're decelerating it's this equation so I hate that I really do so we're going to make a point of making you know of making terms positive negative a lot of times and stuff like that throughout this course especially in kinematics and as a result I just find no need to include this equation giving students an extra equation often adds to the confusion making them realize there's only four uniform acceleration equations usually puts a smile on students faces so I'm not going to use this equation so what I'm going to do is is input an acceleration that's either positive or negative right there and again if you're speeding up then the acceleration of the Velocity are in the same direction I'll make it positive if you're slowing down decelerating then the acceleration of velocity point in opposite directions and I'll make the acceleration negative in that case all right they're also not going to show where this one's dry from this is my least favorite of the equations so it's my least favorite because it has squares in it and that makes it a little difficult sometimes to do the math in your head and I like doing math in my head when I can but definitely we'll plug and chug with a calculator when appropriate and things of A Sort uh but one thing that's nice about this equation right here is this is the only one of all the equations here that does not have time in it notice this one has time in it this one has time in it this one has time in it this is the only one that does not have time in it and so if you're doing a calculation involving uniform acceleration but you don't know anything about the time that might be the indication that this is the lovely equation you're using and then finally this last one here this last one just like these first couple were a rearrangement of the definition of velocity this last one is really just a rearrangement of the definition of acceleration so if you recall acceleration was equal to the change in velocity over the change in time and so what you might see is that acceleration equals V final minus V initial all over and again it's really T final minus t initial but if T initial is zero then it's really just T final but if there's only a final if there's only one time point we'll just make it t so at that point so it's like T minus zero and then rearranging it we get a t equals V final minus V initial or V final equals V initial plus a t adding it to the other side and notice that looks just like that in Reverse so that's just really a rearrangement of the definite definition of acceleration and in fact if the numbers are nice I would recommend that if you get an intuitive feel you might not even need this equation because you understand what acceleration actually means and we'll demonstrate this throughout this lesson all right so now we've seen these equations again the first thing you're going to do when approaching any kinematics problem so is first you're going to ask yourself a question is no acceleration I.E constant velocity or is it under uniform acceleration and the problem is going to have to tell you and clue you in in one way shape or form and usually it's going to be by directly telling you which of those two scenarios you're in let's do some plug it in chug it all right once again if you're joining me for my master course all these problems are already typed out for you on the study guides and there's plenty of room so you can work them out right on there if you so choose so but for the rest of you I will make sure that the questions are put up on the screen as we go here all right so first question here a car travels due north for four hours with a constant speed of 60 miles per hour what is the displacement again for those of us that are in the states that's the majority of my audience I'm using units you're going to be somewhat more familiar with to start but definitely be transitioning over to SI units in a little bit so in this case travels due north for four hours with a constant speed the moment you hear constant speed you think constant velocity no acceleration that's the only equation we got all right we want to know what is the total displacement well if this is the only equation we got and it's an equation that involves displacement and that's the one we're gonna have to use and so in this case we've just got the displacement equals the velocity times the time and we'll just do some plugging and chugging here velocity was given as 60 miles per hour and the time was given four hours I didn't make a sure it was in hours that the units would cancel here so and 60 times four is 240 miles so uh and there's a good chance you probably did this one in your head without like plugging and chugging going through an equation like Well Chad the car goes 60 miles per hour that means every hour it goes 60 miles so if you did that for one one hour that would be 60 miles if you did it for two hours that'd be 120 miles three hours 180 miles four hours 240 miles and if you did it that way and that's the level of an intuitive understanding you have fantastic so if you didn't hopefully that last exercise of one two three four hours how far it went uh was instructive but the answer definitely is the displacement is 240 miles all right second problem we'll take a look at this one's going to be a little bit tricky but it says a man travels 60 miles to work one way at an average speed of 40 miles per hour he travels 60 miles home at an average speed of 60 miles per hour what is the average speed of his round trip now you might be tend to be like well 40 miles per hour one way 60 miles per hour the other way the average is 50 right Chad wrong it turns out but we'll see why it is a little bit tricky this makes a great multiple choice question on a test because they can put that 50 miles per hour is definitely one of the answer choices and you might be tempted to just pick it without even trying to work the problem out but it's not going to work out to be 50 as we'll see let's see if we can map this out a little bit so we've got home and over here we've got work and so on his way to work so 60 miles to work so that is a displacement or distance technically but displacement of 60 miles and we're told is average velocity is 40 miles per hour all right then on the way home he's just retracing his steps it's 60 miles back home so again that's a displacement of 60 miles but now maybe he's avoiding rush hour or something his average velocity is now 60 miles per hour okay so again your intuitive your intuition if it's telling you the average is going to be 50 again it kind of looks that way you're like well it's 40 miles an hour on the way there 60 miles an hour on the way back the average 50. except the problem is he doesn't spend the same amount of time going those two average speeds so if you look and you say does he get to work faster or does he get home faster well he's you know traveling faster on the way home so it should take him less time to get home and that's the key here he's going to spend less time averaging 60 miles an hour than he does averaging 40 miles an hour and because he spends more time at an average of 40 miles per hour the overall average for the entire round trip is going to come out closer to 40 than it is to 60. so 50 is right in the middle it's going to be lower than 50 miles per hour that's what your intuition hopefully in the future will start to tell you all right so we got to work this out here uh in this case whether we've got acceleration here or not we're not really given any indication so but we're given average velocities and given average velocities that's fantastic we can go straight here and that way whether he starts and stops and speeds up and slows down all the way to work it doesn't matter we've got the only thing we need is that average velocity we can use Delta x equals V average t and for all we know maybe he just went constantly 40 miles per hour out of the way work and never stop there's no lights no nothing whatever the key is once we know an average velocity though it doesn't matter and we can just go straight to displacement equals average velocity times time all right so what we want to do here is again get the average velocity for his entire round-trip commute and so let's put that up on the board here so the round trip so and for the round trip Delta x equals V average T so if you rearrange that that average velocity is going to equal Delta X over t where we need to know the displacement for the entire round trip and the time for the entire round trip well one of those is not so bad to figure out if it's 60 miles to work and 60 miles home well then this entire displacement for the entire round trip I was tired distance really we should say is 120 miles this displacement would be zero that's tricky so 120 miles for the distance notice really dealing with distance and speed scalars rather than velocity and displacement here all right so 120 miles is a total distance traveled but the time we don't have that so we've got to find the time for on the way to work and the time for on the way home and add them together to get there and to do that we're actually going to use the same set of equations here so on the way to work we've got Delta x equals V average T if you rearrange and solve for T you're going to get Delta X let's see if I can write that correctly over V average which in this case was 60 miles over 40 miles per hour is going to come out to 60 over 40 the same thing as 6 over 4 which reduces down to 3 over 2 or 1.5 hours all right going home this one you've probably seen your head if he's traveling it's an average of 60 miles per hour then how long does it take him to go 60 miles well one hour but again you could set it up just the same way and just say time is equal to the distance over the speed the average speed which is 60 miles over 60 miles per hour which is going to get you one hour and then you're simply going to add these together in one and a half hours plus one hour is two and a half hours cool and that's we're going to plug in time for the entire time of the round trip and so here we've got 120 miles over two and a half hours and you can definitely plug this in your calculator you could also if you wanted to and I like doing math in my head when I can instead of two and a half I can write this as five halves hours so and dividing by five halves is the same thing as multiplying by two-fifths so five goes into 120 24 times times two it's going to get you 48 miles per hour that is the answer to this question cool and as we pointed out before because he spends more time at 40 miles per hour than he does at 60 miles per hour the overall average velocity for the entire round trip should come out closer to 40 than it does to 60 and indeed that's true all right the next question here is a multi-part question it says a car accelerates uniformly from rust with an acceleration of 10 meters per second squared what is the velocity after six seconds how far does it travel in the first second how far does it travel in the second second how far does it travel in the third second and the idea is that if this car is speeding up then it's going to be going faster and faster and faster and so how far it travels every additional second should be further and further and further so let's see kind of how we approach this here so we're told that the uniform acceleration that acceleration is 10 and notice it centimeters per second squared but you might have intuited that I'm going to write meters per second per second to make that pretty so it means it's speeding up 10 meters per second every second nice round number it's going to make doing this not so bad at so first question we want is what is the velocity after six seconds so and then we want how far does it travel in the first the second and the third second so velocity after six seconds this you can do in your head so the acceleration is 10 meters per second per second so we're told it accelerates from rest which means the initial velocity is zero it's a big keyword from rest means your initial velocity is zero so if you start at zero and you're speeding up 10 meters per second every second then one second after you start you should be moving with a velocity of 10 meters per second after two seconds you'll there's an additional 10 meters per second for that second second now you're up to 20 meters per second for the velocity after three seconds up to 30 meters per second for the velocity after four seconds up to 40 meters per second for the velocity again the key is that the acceleration is 10 meters per second every second per second so every second should go further the velocity goes up by 10 meters per second that's how this works and so you're like well then what is the velocity after six seconds well if you started 0 it should be 60 meters per second and hopefully you can kind of see that but if you understand what acceleration means that it's just the change in the velocity per second and it's telling you that it's changing by 10 meters per second every second then after six seconds it'd be 60 meters per second after 10 seconds would be 100 meters per second after 15 seconds it'd be 150 meters per second and what you may not have realized if you're doing it in your head like that is that you're technically just using that equation without actually having to think about using that equation and if you can think of acceleration like that that's exactly where I want you to get and that's why the problems we're going to do are going to be nice round numbers because if instead I'd said instead of saying the acceleration is 10 meters per second squared what if I'd said the acceleration is 1.436 meters per second squared well now all of a sudden the math is tough to do in your head you're going to put out your calculator so and all of a sudden it's not going to be this kind of intuitive thing that you're probably doing in your head but if we start with nice round numbers then when you do get harder numbers hopefully you've got some of those intuitive Pathways built in and you might still realize you need to pull out your calculator but you might recognize what you need to do in your calculator without even thinking about it as an actual equation so but if we did use that equation we'd say that the final velocity equals the initial velocity of zero plus the acceleration of 10 meters per second per second times the time of six seconds and you can see yep that final velocity after six seconds is going to be 60 meters per second notice these seconds cancel and you'll look for units of meters per second okay so same thing we just intuitively calculated now we use an equation to do it as well now let's go to the other problems here and so the question is how far does it travel in the first second the second second and so we want Delta X for zero two one second or maybe we'll write this a little bit different so T equals zero to t equal one second and then we want the displacement for T equals one second to T equals two seconds so the second second and then we want the displacement for T equals two seconds to T equals three seconds cool now there's actually not just one way of approaching this there's a few different ways a couple at least a couple of different ways and we'll go through a couple of them but we want displacement so you got to say okay well which of these equations have displacement well the first three all have displacement that means you're probably using one of those first three probably not using that last one so at least not directly or initially as we'll see let's kind of take an accounting here and figure out how we might do this well in this case first equation Delta x equals V average T let's write that out well for that first second how long a period of time is passing well one second okay we got the time do we have the average velocity during that for a second well we don't but you might recall again that your average velocity is equal to V initial plus v final over two well for that first second at Time Zero what was the velocity well he started from rest the velocity was Zero and after one second we haven't figured that out yet formally but we did informally we said the acceleration is 10 meters per second per second so after one second we said yeah the velocity at that point would be 10 meters per second after two seconds it would be 20 meters per second after three seconds it would be 30 meters per second and so even though we haven't formally done it we can calculate it or just reason it out in our heads based on knowing what acceleration is or use this equation to figure it out and say that V final equals the initial velocity of zero plus acceleration of 10 meters per second per second times one second and you're going to get a velocity of 10 meters per second so even though we're not given what that final velocity is at T equals one second we can Intuit it or calculate it very easily and so yeah we don't know the average velocity off the topper heads but we can get it pretty quickly and totally use this so the initial velocity was Zero the final velocity at T equals one second is 10 meters per second so the average is going to come up to 5 meters per second and from there and again from zero to one is one second and so five meters per second times one second gonna get us five meters so from T equals zero to T equals one second the displacement is five meters so what about for the second second from T equals one second to T equals two seconds how far now well again this whatever this object is it is speeding up and going faster and faster and faster which means every successive second we should anticipate that it's going further and further and further now we don't know the answer yet but it's got to be higher than five meters right let's figure this out we could do this the same way and we could rationalize that on our head so at T equals one second we already figured out that at T equals one second the velocity is 10 meters per second at T equals two seconds we're going to be up to a velocity of 20 meters per second and if you know the initial and final velocities then the average is halfway in between or you can do 10 plus 20 over 2. so and the average would come out to 15 meters per second and we could do the same thing and say the displacement equals the average velocity 15 meters per second and again from T equals one it's equals two as a total duration of time of one second so 15 meter second times one second is 15 meters and we haven't written anything on the board yet we just kind of rationalized that out and did it in our head using again the exact same equation okay now that's not the only way to do it but we could have done it that way and it's 15 meters it's longer than 5 meters just like we expected it to be so but let's approach this a little bit different way let's see if we can go about this route right here and so in this case Delta x equals V initial t plus one half a t squared okay so in this case v initial well V initial at T equals one second is 10 meters per second I'm sorry yes 10 meters per second how long is the entire duration of this journey from T equals one second to equals took it total of one second what's our acceleration 10 meters per second per second and again how long is the entire duration of this part of the journey again it's a full one second long Square it so by the way big mistake students make is forgetting the square terms not just here but another place like equals mc squared that's another common place where students forget to actually Square it they might write it in the equation then forget to do it in the math so this is a key common place where that happens here in physics so let's look at this out so notice 10 meters per second times one second that's 10 meters and then 10 meters per second per second times one second squared is also going to come out to 10 meters then times a half is 5 meters 10 meters plus 5 meters gets us one other way of getting us our 15 meters so more than one way to skin a cat so to speak all right last one here from two seconds to three seconds well from two seconds to three seconds we can see again that at T equals two seconds our velocity equals 20 meters per second by the time we get to T equals three seconds it'll have gone up an additional 10 meters per second based on the acceleration and should be up to 30. meters per second which means the average velocity during that one second duration somewhere exactly right halfway in between of 25 meters per second so and then the whole time is one second so you say Okay average velocity is 25 meters per second times one second and it should be 25 meters and again we just did it in our head or we could use this equation again and say okay the initial velocity is 20 meters per second times again the whole duration from two seconds to three seconds is one second so we got 20 meters per second times one second plus one half times 10 meters per second per second times one second squared and you're going to get 20 plus 5 and get 25 meters per second yet again but maybe there's another way we can do this because displacement also shows up in this equation right here now again this is my least favorite equation I would never use this one for this calculation I just want to demonstrate that it will lead us to the same answer okay so in this case we've got V final squared equals V initial squared plus two a Delta X so and this one is kind of the most laborious because again we've already figured it out that the time at T equals two seconds the initial velocity for this period is 20 meter second the final is 30 meter second but we had to figure those out and they're both going to show up here that we would have to figure those out in this one as well so alien will get a square in which makes the math big and stuff and that's not very fun but that final velocity was 30 meters per second and so here we plug that in 30 meters per second and square it and in fact I'm going to subtract off this initial velocity squared as well and it's 20 meters per second and then we'll square it and that's going to equal our two times our acceleration 10 meters per second per second foreign times Delta X here well 30 squared is 920 squared is 400 900 minus 400 is 500. and then 500 divided by 20 which I might do differently I might do 500 divided by the 10. so is 50 and then divided by the 2 is 25 and yet again I get Delta x equals 25 meters get the same answer anyway here now it won't always work out like this like oh you just pick any other questions it worked out in this one that we could have used any one of those three equations but which one's the easiest and personally for me the easiest one for me was again that guy right there I could pretty quickly knowing what acceleration actually means figure out that the initial velocity at T equals two seconds so it's 20 meters per second and at T equals three seconds was 30 meters per second and at the average is going to be halfway in between at 25 meters per second and so during that one second period 25 meters per second times a second is 25 meters that was definitely the easiest way for me but all three of those equations could have worked