In previous videos, we looked at motion along a straight line, and now we're going to look at motion
along a curved line, now curved line can be in two dimensions or it could be in three dimensions to
generalize.
We're going to look at motion along a curved line in three dimensions, which means you need three coordinates
to describe this motion.
So the way we do this is we define an origin and say that is the origin there and it's got coordinates
zero zero zero and we have a particle that's going to move along some curved path, a certain distance
from the origin.
So this origin is point zero zero zero.
And we're going to have a particle that moves along this curved line.
It said that's our particle.
Now, this particle is going to be a distance or from the origin.
And that or is a vector, it's got three coordinates indicating the position of this particle, this
particle can move and let's say this particle moves to here in a certain time and you can see that or
is going to change.
So the coordinates of or is going to change, let's call that or Brawne.
And the direction of the velocity of this particle is also going to change in this case, the velocity
is in the direction of a year, the velocity has changed.
It's more in that direction.
Again, this we need to describe.
So, again, we're going to look at position, velocity and acceleration.
Position we described by our.
Position, position or.
Is a certain amount X.
In the eye direction unit, Victor, I certain in the X direction plus a certain amount in the direction
plus a certain amount in the key direction.
Now I.
J and those are unit vectors in the X, Y and Z axis directions.
So this is how we describe our position on if you work on a flat plane, like on the page of the there
is no Z coordinates, it's only a certain distance along the X axis in the other direction and a certain
distance along the Y axis and the Z in the direction.
But I keep on adding the K direction because that is totally acceptable, that this particle can move
in three dimensions.
Now, here's the thing.
If we take the time derivative of position, what do we get?
We get velocity.
So velocity.
Velocity V and I add a little line there, because it is now a vector in three dimensions, three dimensions.
It's going to be still the answer to it.
Previously it was the HST, we just signify position with an Aurier when when we're not along a straight
line, but along a curve path.
And so that is just going to be the components, it's going to be the X, the T in the other direction,
so there's a certain velocity in this direction.
Plus the Y did it in the J direction, plus the Z, T in the K direction.
And we can give an example of this is an example, example would be that our velocity of a certain particle,
let's say this particle at this point has a certain velocity and that velocity could be something like,
let's say three T.
In the other direction, plus five T in the J direction plus two K.
Plus two in the key direction.
Sorry about that, so you can see it is a function of time.
It doesn't necessarily have to be a function of time.
It could always be, too, in the direction you of a team there.
But the point I'm trying to make is that it is a function in three different directions.
And we sort of have three independent equations, three independent components for the velocity in each
one of the three principal directions, the I, J and K Direxion.
So now the next one is acceleration.
So.
Let's look at acceleration.
We know that acceleration is the term derivative of velocity acceleration, a time derivative of the
velocity, sorry, my Vee's looked like an R sometimes, but it's a V.
So that would be what is V. We have a we know that V is the first derivative of position in each direction.
So we're going to have the derivative of so we're going to have Divi Deti, D.V. X, which is the X
component of the velocity in the eye unit vector direction, plus the the Y T in the general direction
plus the V Z, T in the K direction.
So once again we have different components.
Of the acceleration for each one of our directions age, OK?
Now, what is the absolute acceleration, the absolute exhilaration, it's going to be the magnitude
of these components and the way we calculate that is that we take the square root of each of the components.
So that would be the square root.
Now, instead of writing all of this, I'm just going to write a X, so we X.
Squared plus a Y squared, there's a Z squared.
So I take each one of these acceleration components, square them, take the square root to find the
magnitude.