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  • What I want to do with this video

  • is think about what happens to some type of projectile,

  • maybe a ball or rock, if I were to throw it

  • straight up into the air.

  • To do that I want to plot distance relative to time.

  • There are a few things I am going to tell you about

  • my throwing the rock into the air.

  • The rock will have an initial velocity (Vi) of 19.6 meters per second (19.6m/s)

  • I picked this initial velocity because

  • it will make the math a little bit easier.

  • We also know the acceleration near the surface of the earth.

  • We know the force of gravity

  • near the surface of the earth is the mass of the object times the acceleration.

  • (let me write this down)

  • The force of gravity is going to be the mass of the object times little g.

  • little g is gravity near the surface of the earth

  • g is 9.8 meters per second squared (9.8m/s^2)

  • Now if you want the acceleration on earth

  • you just take the force divided by the mass

  • Because we have the general equation

  • Force equals mass times acceleration (F=ma)

  • If you want acceleration divide both sides by mass

  • so you get force over mass

  • So, lets just divide this by mass

  • If you divide both sides by mass,

  • on the left hand side you will get acceleration

  • and on the right hand side you will get the quantity little g.

  • The whole reason why I did this is

  • when we look at the g it really comes from

  • the universal law of gravitation.

  • You can really view g as

  • measuring the gravitational field strength near the surface of the earth.

  • Then that helps us figure out the force

  • when you multiply mass times g.

  • Then you use F=ma, the second law,

  • to come up with g again

  • which is actually the acceleration.

  • This is accelerating you towards the center of the earth.

  • The other thing I want to make clear:

  • when you talk about the Force of gravity

  • generally the force of gravity is equal to big G

  • Big G (which is different than little g) times

  • the product of the masses of the two things

  • over the square of the distance between the two things.

  • You might be saying "Wait, clearly the force of gravity is dependent on the distance.

  • So if I were to throw something up into the air,

  • won't the distance change."

  • And you would be right!

  • That is technically right, but

  • the reality is that when you

  • throw something up into the air

  • that change in distance is so small

  • relative to the distance between the object and the center of the earth

  • that to make the math simple,

  • When we are at or near the surface of the earth (including in our atmosphere)

  • we can assume that it is constant.

  • Remember that little g over there is

  • all of these terms combined.

  • If we assume that mass one (m1) is

  • the mass of the earth, and

  • r is the radius of the earth (the distance from the center of the earth)

  • So you would be correct in thinking that it changes a little bit.

  • The force of gravity changes a little bit, but

  • for the sake of throwing things up into our atmosphere

  • we can assume that it is constant.

  • And if we were to calculate it

  • it is 9.8 meters per second squared

  • and I have rounded here to the nearest tenth.

  • I want to be clear these are vector quantities.

  • When we start throwing things up into the air

  • the convention is

  • if something is moving up it is given a positive value,

  • and if it is moving down we give it a negative value.

  • Well, for an object that is in free fall

  • gravity would be accelerating it downwards, or

  • the force of gravity is downwards.

  • So, little g over here,

  • if you want to give it its direction,

  • is negative. Little g is -9.8m/s2.

  • So, we have the acceleration due to gravity.

  • The acceleration due to gravity (ag) is

  • negative 9.8 meters per second squared (9.8m/s^2).

  • Now I want to plot distance relative to time.

  • Let's think about how we can set up a formula,

  • derive a formula that, if we input time as a variable,

  • we can get distance.

  • We can assume these values right over here.

  • Well actually I want to plot displacement over time because that will be more interesting.

  • We know that displacement is

  • the same thing as average velocity times change in time (displacement=Vavg*(t1-t2)).

  • Right now we have

  • something in terms of time, distance, and average velocity

  • but not in terms of initial velocity and acceleration.

  • We know that average velocity is the same thing as

  • initial velocity (vi) plus final velocity (vf) over 2. (Vavg=(vi+vf)/2)

  • If we assume constant acceleration.

  • We can only calculate Vavg this way assuming constant acceleration.

  • Once again when were are dealing

  • with objects not too far from the center of the earth

  • we can make that assumption.

  • Assuming that we have a constant acceleration

  • Once again we don't have what our final velocity is.

  • So, we need to think about this a little more.

  • We can express our final velocity in terms of our initial velocity and time.

  • Just dealing with this part, the average velocity.

  • So we can rewrite this expression

  • as the initial velocity plus something over 2.

  • and what is final velocity?

  • Well the final velocity is going to be

  • your initial velocity plus your acceleration times change in time.

  • If you are starting at 10m/s

  • and you are accelerated at 1m/s^2

  • then after 1 second you will be going 1 second faster than that. (11m/s)

  • So this right here is your final velocity.

  • Let me make sure that these are all vector quantities...(draws vector arrows)

  • All of these are vector quantities.

  • Hopefully it is ingrained in you that these are all vector quantities, direction matters.

  • And let's see how we can simplify this

  • Well these two terms

  • (remember we are just dealing with the average velocity here)

  • These two terms if you combine them become 2 times initial velocity (2vi).

  • two times my initial velocity

  • and then divided by this 2

  • plus all of this business divided by this 2.

  • which is my acceleration times my change in time divided by 2.

  • All of this was another way to write average velocity.

  • the whole reason why I did this is because we don't have final velocity

  • but we have acceleration

  • and we are going to use change in time as our independent variable.

  • We still have to multiply this by this green change in time here.

  • multiply all of this times the green change in time.

  • All of this is what displacement is going to be.

  • This is displacement, and lets see...

  • we can multiply the change in time times all this

  • actually these 2s cancel out

  • and we get (continued over here)

  • We get: displacement is equal to

  • initial velocity times

  • change in time

  • Some physics classes or textbooks put time there but it is really change in time.

  • change in time is a little more accurate

  • plus 1/2 (which is the same as dividing by 2)

  • plus one half times the acceleration

  • times the acceleration

  • times (we have a delta t times delta t)

  • change in time times change in time

  • the triangle is delta and it just means "change in"

  • so change in time times change in time

  • is just change in times squared.

  • In some classes you will see this written as

  • d is equal to vi times t plus 1/2 a t squared

  • this is the same exact thing

  • they are just using d for displacement

  • and t in place of delta t.

  • The one thing I want you to realize with this video

  • is that this is a very straight forward thing to derive.

  • Maybe if you were under time pressure you would want to be able to whip this out,

  • but the important thing, so you remember how to do this when you are 30 or 40 or 50

  • or when you are an engineer and you are trying

  • to send a rocket into space and you don't have a physics book to look it up,

  • is that it comes from the simple displacement is equal to average velocity times change in time

  • and we assume constant acceleration,

  • and you can just derive the rest of this.

  • I am going to leave you there in this video.

  • Let me erase this part right over here.

  • We are going to leave it right over here.

  • In the next video we are going to use this

  • formula we just derived.

  • We are going to use this to actually

  • plot the displacement vs time

  • because that is interesting and we are going to be thinking about

  • what happens to the velocity and the acceleration

  • as we move further and further in time.

What I want to do with this video

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B1 中級

推導位移作為時間、加速度和初速度的函數|可汗學院 (Deriving displacement as a function of time, acceleration, and initial velocity | Khan Academy)

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    楊凱翔 發佈於 2021 年 01 月 14 日
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