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# Average velocity for constant acceleration

Calculating average velocity when acceleration is constant. Created by Sal Khan.

Video transcript

The goal of this video is to explore some of the concept of formula you might see in introductional physics class but more importantly to see they are really just common sense ideas So let's just start with a simple example Let's say that and for the sake of this video keep things that magnitudes and velocities that's the direction of velocity etc. let's just assume that if I have a positive number that it means for example postive velocity that it means I'm going to the right let's say I have a negative number we won't see in this video let's assume we are going to the left In that way I can just write a number down only operating in one dimension you know that by specifying the magnitude and the direction if I say velocity is 5m/s that means 5m/s to the right if I say negative 5m/s that means 5m/s to the left let's say just for simplicitiy, say that we start with initial velocity we start with an initial velocity of 5m/s once again I specify the magnitude and the direction because of this convention here, we know it is to the right let's say we have a constant acceleration we have a constant acceleration 2m/s^2 or 2m per second square 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, let's say we do this for a duration of 4 I will just use s, second and s different places so s for this video is seconds So I want to do is to think about how far do we travel? and there is two things how fast are we going? after 4 seconds and how far have we travel over the course of those 4 seconds? so let's draw ourselves a little diagram here So this is my velocity axis, and this over here is time axis we have to draw a straighter line than that So that is my time axis, time this is velocity This is my velocity right over there and I'm starting off with 5m/s, so this is 5m/s right over here So vi is equal to 5m/s And every second goes by it goes 2m/s faster that's 2m/s*s every second that goes by So after 1 second when it goes 2m/s faster it will be at 7 another way to think about it is the slope of this velocity line is my constant accleration, my constant slope here so it might look something like that So what has happend after 4s? So 1 2 3 4 this is my delta t So my final velocity is going to be right over there I'm writing it here because this get into the way of veloctiy so this is v this is my final velocity what would it be? Well I'm starting at 5m/s So we are doing this both using the variable and concretes Some starting with some initial velocity I'm starting with some initial velocity Subscript i said i for initial and then each second that goes by I'm getting this much faster so if I gonna see how much faster have I gone I multiply the number of second, I will just multiply the number second it goes by times my acceleration, times my acceleration and once again, this right here, subscript c saying that is a constant acceleration, so that will tell my how fast I have gone If I started at this point and multiply the duration time with slope I will get this high, I will get to my final velocity just to make it clear with the numbers, this number can really be anything I'm just taking this to make it concrete in your mind you have 5m/s plus 4s plus, I wanna do it in yellow plus 4s times our acceleration with 2m per second square and what is this going to be equal to? you have a second that is cancelling out one of the second down here You have 4 times, so you have 5m/s plus 4 times 2 is 8 this second gone, we just have 8m/s or this is the same thing as 13m/s which is going to be our final velocity and I wanna take a pause here, you can pause and think about it yourself this whole should be intuitive, we are starting by going with 5m/s every second goes by we are gonna going 2m/s faster so after 1s it would be 7 m/s, after 2s we will be 9m/s after 3s we will be 11m/s, and after 4s we will be at 13 m/s so you multiply how much time pass times acceleration this is how much faster we are gonna be going, we are already going 5m/s 5 plus how much faster? 13 m/s so this right up here is 13m/s So I will take a little pause here hopefully intuitive and the whole play of that is to show you this formula you will see in many physics book is not something that randomly pop out of there it just make complete common sense Now the next thing I wanna talk about is what is the total distance that we would have travel? and we know from the last video that distance is just the area under this curve right over here, so it's just the area under this curve you see this is kind of a strange shape here how do I caculate this area? and we can use a little symbol of geometry to break it down into two different areas, it's very easy to calculate their areas two simple shapes, you can break it down to two, blue part is the rectangle right over here, easy to figure out the area of a rectangle and we can break it down to 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 even this will hopefully make some intuition because this blue area is how far we would have travel if we are not accelerated, we just want 5m/s for 4s so you goes 5m/s 1s 2s 3s 4s so you are going from 0 to 4 you change in time is 4s so if you go 5m/s for 4s you are going to go 20 m this right here is 20m that is the area of this right here 5 times 4 this purple or magentic area tells you how furthur than this are you going because you are accelerating because kept going faster and faster and faster it's pretty easy to calculate this area the base here is still 5(4) because that's 5(4) second that's gone by what's the height here? The height here is my final velocity minus my initial velocity minus my initial velocity or it's the change in velocity due to the accleration 13 minus 5 is 8 or this 8 right over here it is 8m/s so this height right over here is 8m/s the base over here is 4s that's the time that past what's this area of the triangle? the area of this triangle is one half times the base which is 4s 4s times the height which is 8m/s times 8m/s second cancel out one half time 4 is 2 times 8 is equal to 16m So the total distance we travel is 20 plus 16 is 36m that is the total I could say the total displacement and once again is to the right, since it's positive so that is our displacement What I wanna do is to do the exact the same calculation keep it in variable form, that will give another formula many people often memorize You might understand this is completely intuitive formula and that just come out of the logical flow of reasoning that we went through this video what is the area once again if we just think about the variables? well the area of this rectangle right here is our initial velocity times our change in time, times our change in time So that is the blue rectangle right over here, and 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 our final velocity minus our initial velocity these are all vectors, they are just positive if going to the right we just multiply the base with the height that will just be the area of the entire rectangle I will take it by half because triangle is just half of that rectangle so times one half, so times one half so this is the area, 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 the delta t, so you factor out the delta t you get delta t times a bunch of stuff v sub i your initial velocity we factor this out plus this stuff, plus this thing right over here and we can distribute the one half we factor the one half, we factor the delta t out, taking it out and let's multiply one half by each of these things so it's gonna be plus one half times vf, times our final velocity that's not the right color, I will use the right color so you would understand what I am doing, so this is the one half so plus one half times our final velocity final velocity minus one half, minus one half times our initial velocity I'm gonna do that in blue, sorry I have trouble in changing color today minus one half times our initial velocity, times our initial velocity and what is this simplify do? we have something plus one half times something else minus one half of the original something so what is vi minus one half vi? so anything minus its half is just a positive half left so these two terms, this term and this term will simplify to one half vi one half initial velocity plus one half times the final velocity plus one half times the final velocity and all of that is being multiplied with the change in time the time that has gone by and this tells us the distance, the distance that we travel another way to think about it, let's factor out this one half you get distance that is equal to change in time times factoring out the one half vi plus vf, vi no that's not the right color vi plus vf so this is interesting, the distance we travel is equal to one half of the initial velocity plus the final velocity so this is really if you just took this quantity right over here it's just the arithmetic, I have trouble saying that word it's the arithmetic mean of these two numbers, so I'm gonna define, this is something new, I'm gonna call this the average velocity we have to be very careful with this this right here is the average velocity but the only reason why I can just take the starting velocity and ending velocity and adding them together and divide them by two since you took an average of two thing it's some place over here and I take that as average velocity it's because my acceleration is constant which is usually an assumption in introductory physics class but it's not always the assumption but if you do have a constant acceleration like this you can assume that the average velocity is gonna be the average of the initial velocity and the final velocity if this is a curve and the acceleration is changing you could not do that but what is useful about this is if you wanna figure out the distance that was travelled, you just need to know the initial velocity and the final velocity, average their two and multiply the times it goes by so in this situation our final velocity is 13m/s our initial velocity is 5m/s so you have 13 plus 5 is equal to 18 you divided that by 2, you average velocity is 9m/s if you take the average of 13 and 5 and 9m/s times 4s gives you 36m so hopefully it doesn't confuse you I just wannt show you some of these things you will see in your physics class but you shouldn't memorize they can all be deduced