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  • 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

The goal of this video is to explore some of the concept of formula you

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恆定加速度的平均速度|一維運動|物理學|可汗學院 (Average velocity for constant acceleration | One-dimensional motion | Physics | Khan Academy)

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    楊凱翔 發佈於 2021 年 01 月 14 日
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