Welcome back.
I will now introduce you to the concept of-- well, I've
introduced what vectors are, but I'll introduce you to how
they can be even more useful and we're going to use that to
actually solve 2-dimensional projectile motion problems.
Now we can introduce the game where you just don't throw the
ball straight up, you can see how far you can throw the ball
or what's the optimal angle to throw the ball at.
So hopefully, you'll find this even more fun, if you can
imagine it, then the
1-dimensional projectile motion.
So everything we've been doing so far, hopefully, you have an
intuition that the difference between a vector and a scalar
is that a vector has direction.
So a vector has-- let me do colon instead-- has magnitude
and direction.
While a scalar has only magnitude.
An example of a vector, velocity is a vector.
That's easy.
Maybe we'll start with v, but velocity is a vector.
Well in those examples we were doing before, we said that the
ball was going-- sometimes we said that the final velocity
was minus 30 meters per second.
And so where was the magnitude and what was the direction?
Well, the magnitude was 30.
And the direction was-- I knew what the direction was by the
sign of that minus sign.
In that same problem, if someone asked me, what is the
speed of the ball right before it hits the ground?
I would've said oh, the speed is just 30 meters per second.
And I wouldn't have to say 30 meters per second down or
minus 30 per second.
I'd just say 30 meters per second
because that's its speed.
I don't know what direction it's going in that speed.
So that's its speed.
And in a general way, in one dimension, you could say that
the scalar of the same quantity tends to be the
absolute value of the vector quantity.
Because in this example the speed is just the absolute
value of this.
If my initial velocity was plus-- that's not equal,
that's actually just equal.
If the initial velocity was plus 30 meters per second,
then the speed would still be 30 meters per second.
So I just wanted to introduce that notion.
And in one dimension, velocity can kind of be-- direction can
be implied by positive or negative numbers.
But now, when we go into two dimensions, we're going to
have to get a little bit more sophisticated in terms of how
we give direction.
So let me clear this.
Let me give an example where I take a ball.
I take a green ball, clearly a green ball, and I throw it--
this is a straight line.
I throw it at 10 meters per second, but I don't throw it
straight up.
I actually throw it at a-- let me think of a good angle.
I throw it at a-- I don't know.
I throw it at a 45 degree angle.
Or if you like, radians, that's what?
So that's pi, pi over 4.
Let's say I throw it at a 45 degree angle.
So now my problem becomes a lot more complicated.
One, how do I specify the magnitude and direction?
And then, how do I solve this problem?
How do I figure out how high it goes, how far it goes.
et cetera.
And hopefully, that's what we will be able to teach you over
the course of the next couple of videos.
This is the ground, right?
So one, if I just want to give the speed and direction, or
the magnitude and the direction for this vector
quantity, I have just specified it.
You know, I could to say my velocity or my initial
velocity is 10 meters per second at 45 degrees to the
horizon or something like that.
So that would be a legitimate velocity specification.
But it's kind of hard to deal with, so what I'm going to
teach you is something very neat about vectors.
Is that you can actually break up a vector into-- I don't
want to introduce two complicated terms, but into
its orthogonal components.
So you can say well, something moving at 10 meters per second
at this angle, is actually moving at some speed upwards
and it's moving at some speed, in this case, to the right.
Let's say in the positive direction.
So if you imagine this coordinate axis, it's moving
in the y direction at some speed and it's moving in the x
direction at some speed.
And when they're combined at some way, you get this angle
or you're moving at kind of an incline.
So how do I figure out-- if I have 10 meters per second and
I'm moving-- how do I break that down into my vertical
speed and my horizontal speed?
And I'll show you that it's actually pretty cool.
It's pretty straightforward.
What you can do is you can actually draw a vector where
the length of the vector is its magnitude.
Let's draw it again.
So let's say this is a coordinate axis and that the
length of this line right here is 10.
The length of the this line is 10 and it represents speed or
velocity at this point.
And we know this angle is 45 degrees.
So it turns out that the vertical component of the
ball's velocity would be equivalent to the length of
this side, and the horizontal component would be equivalent
to the length of this side.
This arrow right here is the same thing as this arrow, I
just translated it to the right.
Just so that you see that it's just a triangle.
And it's just not any triangle,
it's a right triangle.
And what's useful-- I don't want to get too complicated
with linear algebra.
But it's important that the two directions that you're
breaking it up in are-- well, they don't have to be, but it
makes it useful that they are at right angles to each other
because then we can start to use a lot of our trig ratios.
So let's just figure out this problem.
I think you just need to see a lot of example.
So we're going 10 meters per second at 45 degrees.
Well, what speed am I going straight up then?
What is the straight up component of this 10 meters
per second?
Well, how do we figure out this side if this was just a
regular triangle?
If this is the hypotenuse, this is
opposite and this is adjacent.
So let me break out some trigonometry.
soh cah toa.
And if you don't remember your trig or you never learned
trig, I encourage you to go watch the trig videos that we
have. soh cah toa.
Which of the trig ratios involve the opposite and the
hypotenuse?
So our sine.
And we know that sine of theta is equal to opposite over
hypotenuse.
So we know that the sine of 45 degrees is equal to-- let's
call this my velocity of vertical.
I've never seen that variable used in actual physics
classes, but I think it makes sense.
This is the velocity vertical and this is the velocity
horizontal.
And later I'll introduce more conventional conventions, but
that'll do us for now.
So my vertical velocity, the velocity that I'm going
straight up is-- well, sine of 45 degrees is my vertical
velocity over my kind of angled velocity, this
hypotenuse over 10.
And what's my horizontal velocity?
And so if I were to solve for vertical velocity, I'd get
vertical velocity is equal to 10 times
the sine of 45 degrees.
And then similarly, my horizontal velocity, this is
the adjacent over hypotenuse-- cah.
So it's cosine.
So I could say cosine of 45 degrees is equal to my
horizontal velocity over 10.
Or my horizontal velocity is equal to 10 times the cosine
of 45 degrees.
I'm about to run out of time, so I'm going to continue this
in the next video.