Two-dimensional projectile motion
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Visualizing Vectors in 2 Dimensions
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Projectile at an Angle
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Different Way to Determine Time in Air
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Launching and Landing on Different Elevations
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Total Displacement for Projectile
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Total Final Velocity for Projectile
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Correction to Total Final Velocity for Projectile
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Projectile on an Incline
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Unit Vectors and Engineering Notation
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Clearing the Green Monster at Fenway
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Green Monster at Fenway Part 2
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Unit Vector Notation
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Unit Vector Notation (part 2)
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Projectile Motion with Ordered Set Notation
Unit Vector Notation Expressing a vector as the scaled sum of unit vectors
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- Good afternoon.
- We've done a lot of work with vectors.
- In a lot of the problems, when we launch something into—In
- the projectile motion problems, or when you were
- doing the incline plane problems. I always gave you a
- vector, like I would draw a vector like this.
- I would say something has a velocity of
- 10 meters per second.
- It's at a 30 degree angle.
- And then I would break it up into the x and y components.
- So if I called this vector v, I would use a notation, v sub
- x, and the v sub x would have been this vector right here.
- v sub x would've been this vector down here.
- The x component of the vector.
- And then v sub y would have been the y component of the
- vector, and it would have been this vector.
- So this was v sub x, this was v sub y.
- And hopefully by now, it's second nature of how we would
- figure these things out. v sub x would be 10 times cosine of
- this angle.
- 10 cosine of 30 degrees, which I think is square root of 3/2,
- but we're not worried about that right now.
- And v sub y would be 10 times the sine of that angle.
- This hopefully should be second nature to you.
- If it's not, you can just go through SOH-CAH-TOA and say,
- well, the sine of 30 degrees is the opposite of the
- hypotenuse.
- And you would get back to this.
- But we've reviewed all of that, and you should review
- the initial vector videos.
- But what I want you to do now, because this is useful for
- simple projectile motion problems-- But once we start
- dealing with more complicated vectors-- and maybe we're
- dealing with multi-dimensional of vectors, three-dimensional
- vectors, or we start doing linear algebra, where we do
- n-dimensional vectors—we need a coherent way, an
- analytical way, instead of having to always draw a
- picture of representing vectors.
- So what we do is, we use something I call, and I think
- everyone calls it, unit vector notation.
- So what does that mean?
- So we define these unit vectors.
- Let me draw some axes.
- And it's important to keep in mind, this might seem a little
- confusing at first, but this is no different than what
- we've been doing in our physics problem so far.
- Let me draw the axes right there.
- Let's say that this is 1, this is 0, this is 2.
- 0, 1, 2.
- I don't know if must been writing an Arabic or
- something, going backwards.
- This is 0, 1, 2, that's not 20.
- And then let's say this is 1, this is 2, in the y direction.
- I'm going to define what I call the unit vectors in two
- dimensions.
- So I'm going to first define a vector.
- I'll call this vector i.
- And this is the vector.
- It just goes straight in the x direction, has no y component,
- and it has the magnitude of 1.
- And so this is i.
- We denote the unit vector by putting this little
- cap on top of it.
- There's multiple notations.
- Sometimes in the book, you'll see this i without the cap,
- and it's just boldface.
- There's some other notations.
- But if you see i, and not in the imaginary number sense,
- you should realize that that's the unit vector.
- It has magnitude 1 and it's completely in the x direction.
- And I'm going to define another vector, and that one
- is called j.
- And that is the same thing but in the y direction.
- That is the vector j.
- You put a little cap over it.
- So why did I do this?
- Well, if I'm dealing with two dimensions.
- And as later we'll see in three dimensions, so there
- will actually be a third dimension and we'll call that
- k, but don't worry about that right now.
- But if we're dealing in two dimensions, we can define any
- vector in terms of some sum of these two vectors.
- So how does that work?
- Well, this vector here, let's call it v.
- This vector, v, is the sum of its x
- component plus its y component.
- When you add vectors, you can put them head
- to tail like this.
- And that's the sum.
- So hopefully knowing what we already know, we knew that the
- vector, v, is equal to its x
- component plus its y component.
- When you add vectors, you essentially just put
- them head to tails.
- And then the resulting sum is where you end up.
- It would be if you added this vector, and then you put this
- tail to this head.
- And you end up there.
- So you end up there.
- So that's the vector.
- So can we define v sub x as some multiple of i, of this
- unit vector?
- Well, sure.
- v sub x completely goes in the x direction.
- But it doesn't have a magnitude of 1.
- It has a magnitude of 10 cosine 30 degrees.
- So its magnitude is ten.
- Let me draw the unit vector up here.
- This is the unit vector i.
- It's going to look something like this and this.
- So v sub x is in the exact same direction, and it's just
- a scaled version of this unit vector.
- And what multiple is it of that unit vector?
- Well, the unit vector has a magnitude of 1.
- This has a magnitude of 10 cosine of 30 degrees.
- I think that's like, 5 square roots of 3, or
- something like that.
- So we can write v sub x-- I keep switching colors to keep
- things interesting.
- We can write v sub x is equal to 10 cosine of 30 degrees
- times-- that's the degrees --times the unit vector i--
- let me stay in that color, so you don't confused --times the
- unit vector i.
- Does that make sense?
- Well, the unit vector i goes in the exact same direction.
- But the x component of this vector is just a lot longer.
- It's 10 cosine 30 degrees long.
- And that's equal to-- cosine of 30 degrees is square root
- of 3/2 --so that's 5 square roots of 3 i.
- Similary, we can write the y component of this vector as
- some multiple of j.
- So we could say v sub y, the y component-- Well, what is sine
- of 30 degrees?
- Sine of 30 degrees is 1/2.
- 1/2 times 10, so this is 5.
- So the y component goes completely in the y direction.
- So it's just going to be a multiple of this vector j, of
- the unit vector j.
- And what multiple is it?
- Well, it has length 5, while the unit vector
- has just length 1.
- So it's just 5 times the unit vector j.
- So how can we write vector v?
- Well, we know the vector v is the sum of its x component and
- its y component.
- And we also know, so this is a whole vector v.
- What's its x component?
- Its x component can be written as a multiple
- of the x unit vector.
- That's that right there.
- So you can write it as 5 square roots of 3
- i plus its y component.
- So what's its y component?
- Well, its y component is just a multiple of the y unit
- vector, which is called j, with the little
- funny hat on top.
- And that's just this.
- It's 5 times j.
- So what we've done now, by defining these unit vectors--
- And I can switch this color just so you
- remember this is i.
- This unit vector is this.
- Using unit vectors in two dimensions, and we can
- eventually do them in multiple dimensions, we can
- analytically express any two dimensional vector.
- Instead of having to always draw it like we did before,
- and having to break out its components and
- always do it visually.
- We can stay in analytical mode and non graphical mode.
- And what makes this very useful is that if I can write
- a vector in this format, I can add them and subtract them
- without having to resort to visual means.
- And what do I mean by that?
- So if I had to find some vector a, is equal to, I don't
- know, 2i plus 3j.
- And I have some other vector b.
- This little arrow just means it's a vector.
- Sometimes you'll see it as a whole arrow.
- As, I don't know, 10i plus 2j.
- If I were to say what's the sum of these two
- vectors a plus b?
- Before we had this unit vector notation, we would have to
- draw them, and put them heads to tails.
- And you had to do it visually, and it would take
- you a lot of time.
- But once you have it broken up into the x and y components,
- you can just separately add the x and y components.
- So vector a plus vector b, that's just 2 plus 10 times i
- plus 3 plus 2 times j.
- And that's equal to 12i plus 5j.
- And something you might want to do, maybe I'll do it in the
- future video, is actually draw out these two vectors and add
- them visually.
- And you'll see that you get this exact answer.
- And as we go into further videos, or future videos,
- you'll see how this is super useful once we start doing
- more complicated physics problems, or once we start
- doing physics with calculus.
- Anyway, I'm about to run out of time on the ten minutes.
- So I'll see you in the next video.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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