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# Average velocity for constant acceleration

## Video transcript

the goal of this video is to explore some of the concepts or formulas you might see in a traditional physics class but even more importantly to see that they're really just common-sense ideas so let's just let's just start with a simple example let's say that and for the sake of this video just so I stop don't have to keep saying this is the magnitude of the velocity this is the direction of the velocity etc etc let's just assume that if I have a positive number that it means for example if I have a positive velocity it means that I am going to the right and let's say if I have a negative number which we won't see in this video let's assume that I'm going to the left and that way I can just write a number down we're only operating in one dimension you know that it's specifying both the magnitude and the direction if I say the velocity is five meters per second that means five meters per second to the right if I say it's negative five meters per second that means it's five meters per second to the left now let's just say just for simplicity let's just say that we start with an initial velocity we start with an initial velocity of five meters per second and once again I am specifying about the magnitude and the direction because of this convention here we know it is to the right and let's say that we have a constant acceleration we have a constant acceleration at play of two meters per second per second or two meters per second squared and once again since this is positive it is to the right and let's say that we do this for a duration so my change in time is let's say we do this for a duration of four oh four I'll use the same four I'll just use s so I don't use seconds and then second then s different places so s for this video is seconds so what I want to do is think about how far how far do we travel and well there's two things that how far how fast are we going after four seconds and how far have we traveled over the course of those four seconds so let's draw a little drawer ourselves a little bit of a diagram here so this is my velocity axis and this over here is my time axis I can draw a straighter line than that so that is my time axis time this is velocity it's clipping off the left hand that is my velocity right over there I'm starting off at five meters per second so this is five meters per second right over here so VI is equal to five meters per second and then every second that goes by I'm going to meters per second faster because this is two meters per second per second so every second that goes by so after one second I'm going to go two meters per second faster so I'll be at seven or another way to think about it is the slope of this velocity line is my constant acceleration or I have a constant slope here so it might look something it might look something like that so what has happened after four seconds so one two three four so this is my delta T so my final velocity is going to be right over there so my final velocity is going to be right over there I'll write it down here just because it's getting in the way of this word velocity and so this is V this is my final velocity and what would it be well I'm starting at five meters per second so I'm going to do it both using the variables and using the concrete numbers so I'm starting at some initial velocity I'm starting at some initial velocity the subscript I says I for initial and then each second that goes by I'm getting this much faster so if I want to see how much faster have I gone I multiply the number of seconds I'll just multiply the number of seconds that go by times my acceleration times times my acceleration and once again this right here this is I just wrote the subscript C saying that it's a constant acceleration and so that will tell me how fast I've gone if I take if I start at this point and I multiplied the duration times my slope I will get this hi I will get to my final velocity I will get to my final velocity and just to make it clear what the numbers these numbers really could be anything I'm just picking these to make it concrete in your mind you have five meters per second plus four seconds plus I'm going to do that in yellow plus four seconds four seconds times our acceleration of two meters per second squared two meters per second squared and what is this going to be equal to you have a second canceling out with one of the seconds down here you have four times so let me write it so we have five meters per second plus four times two is eight eight this seconds gone we're just left with meters per second eight meters per second or this is the same thing as 13 meters per second which is going to be our final velocity and I want to take a pause here and you can pause and think about it yourself this hopefully should be intuitive we were starting at going five meters per second every second that goes by we're going to go two meters per second faster so after one second we'll be at 7 meters per second after 2 seconds we will be at 9 meters per second after after 3 seconds we will be at 11 meters per second and then after 4 seconds we will be at 13 meters per second so you multiply how much time pass times acceleration this is how much faster we're going to be going if we're already going 5 meters per second 5 plus how much faster 13 meters per second so this right up here is 13 13 meters per second so I'll take a little pause here hopefully that's intuitive and now and the whole point of that is to show you that this formula that you'll often see in many physics books it's not just something that randomly popped out of there it just makes complete common sense now the next thing I want to talk about is what is the total distance that we would have traveled now we know from the last video that distance is just the area under this curve right over here so it's just the area it's the area under this curve you say well this is kind of a strange shape right here how do I calculate its area and we can just use a little simple geometry to break it down into two different areas that are very easy to calculate their areas or that there are two simple shapes you could break it down into this blue part this rectangle right over here easy to figure out the area of a rectangle and we could break it down into this purple part this triangle right here easy to figure out the area of a triangle and that will be the total distance we travel and even this will hopefully make some intuition because this blue area is how far would we have traveled if we were not accelerating if we just went 5 meters per second for four seconds so if you go five meters per second times so this is one second two second three second four second so you're going from zero set time 0 to time for your change in time is four seconds so if you go five meters per second for four seconds you're going to go 20 meters this right here is 20 meters that is the area of this right here five times for this this this my guess purple or magenta area tells you how further than this are you going because you're accelerating because you kept going faster and faster and faster and it's pretty easy to calculate this area the check the base the base here is still 5 because that's five seconds have gone by and what's the height here the height here is my final velocity minus my initial velocity - my initial velocity or it's the change in velocity due to the acceleration and the change in velocity to do the acceleration 13 minus five is eight or it's this eight right over here it is eight it is eight meters per second so this height right over here is eight meters per second the base right over here is fat and it sorry it's four seconds that's the time that passed so what's the area of this triangle well area of a triangle is one half times the base which is four seconds four seconds times the height which is eight meters per second times eight meters per second seconds cancel out 1/2 times 4 is 2 times 8 is equal to is equal to 16 is equal to 16 meters so the total distance we traveled is 20 plus 16 is 36 meters that is the total or I could say the total displacement and once again it's to the right since it is positive so this is our displacement now what I want to do is do this exact same calculation but keep it in variable form and that'll give another formula that many people many often memorize but I want you to understand it it's a completely intuitive formula and it just comes out of the kind of the the logical flow of reasoning that we went through this video what is the area once again if we use in the if we just think about the variables well the area of this of this rectangle right here is our initial velocity times our change in time times our change in time so that's the blue this is the blue rectangle right over here and then plus what do we have to do we have the change in time once again we have the change in time times this height which is our final velocity which is times our final velocity minus our initial velocity minus our initial velocity these are all vectors oh they're just positive telling us we're going to the right and if we just don't multiply the base times the height that would give us the area of this entire rectangle we have to take it by half because the triangle is only half of that rectangle so times one-half so times so times one-half and so this is the area this is the purple area that's not purple this is the purple area right over here this is the area of this this is the area of that and let's simplify this a little bit let's factor out a delta T so if you factor out a delta T you get delta T times a bunch of stuff V sub I so your initial velocity your initial velocity we factor this out Plus this stuff Plus this thing right over here and we can distribute the 1/2 we factored we factored the 1/2 we Fest sorry we factored the Delta T's out that's what we picking it out and let's multiply the 1/2 times each of these things so it's going to be plus 1/2 times V F times our final velocity that's not the right color let me actually do it in the right color so that you understand what I'm doing so this is the 1/2 so plus 1/2 times our final velocity final velocity minus 1/2 minus 1/2 times our initial velocity minus 1/2 times our initial I want to do that in blue sorry I'm having trouble changing colors today minus 1/2 times our initial velocity times our initial velocity and now what does this simplify to we have something plus 1/2 times something else - 1/2 times that original something so what is VI minus 1/2 times VI so anything minus half of it is you're going to have a positive half left so these two terms this term and this term will simplify to will simplify to 1/2 V I 1/2 the initial velocity plus 1/2 times the final velocity plus 1/2 times the final velocity and all of that is being multiplied by our change in all of that is being multiplied by our change in time or the time that has gone by and this tells us this tells us the distance the distance that we traveled or another way to think about it let's factor out this 1/2 let's factor out this 1/2 you get distance is equal to change in time times factoring out the 1/2 VI plus VF V I know that's not the right color VI plus VF plus VF so this is interesting the distance we traveled is equal to 1/2 of the initial velocity plus the final velocity so this is really if you just took this quantity right over here is just the arithmetic the I always have trouble saying that word it's the arithmetic mean of these two numbers and so I'm going to define this as something new I'm going to call this the average velocity but we have to be very careful with this this right here is the average average velocity but the only reason why I can just take the starting velocity and the ending velocity and adding them together and then divide by 2 is essentially taking the average of these two things which would be someplace over here and I take that as the average velocity is because my acceleration is constant which is usually an assumption in most introductory physics classes but it's not always the assumption but if you do have a constant acceleration like this you can you can assume that the average velocity is going to be the average of the initial velocity and the final velocity if this was a curve if the acceleration was changing you could not do that but what's useful about this is if you want to figure out the distance that was traveled you just need to know the initial velocity and the final velocity average the two and then multiply that times the time that goes by so in this situation our final velocity is 13 meters per second our initial velocity was 5 meters per second so you have 13 plus 5 is equal to 18 you divide that by 2 your average velocity your average velocity is 9 meters per second if you take the average of 13 and 5 and then 9 meters per second times 4 seconds gives you 36 meters so hopefully that doesn't confuse you I just want to show you where some of these things that you'll see in your physics class some of these formulas why you shouldn't memorize them and they can all be deduced