We're now going to move to one of the most prominent examples
of a physics problem in 2 dimensions, and that is projectile motion.
Projectile motion is just the fancy way to say that we throw something
or launched something into the air at some angle from the ground
so that it launches up as you can see here and sort of follows the characteristic of parabolic path.
All kinds of things could be asked about this sort of situation.
But let's first just do a quick overview of the physics, of a scenario like this one,
and what the different variables, variables of motion, in our equations of motions would look like.
First of all it it's in the air and it's near the surface of the earth,
the earth is pulling it down and that's why it's falls.
And so we say that the acceleration of this object because of the pull of the earth,
is g in magnitude, which is 9.8 meter per second squared
and its downwards in direction, and so we give it a negative sign because again,
it is pointing downwards. So in a projectile motion problem near the surface of the earth
until we get into more complicated gravity equations,
near the surface of the earth, you can always assume
that something will have a gravitational acceleration of g,
where 9.8 meters per seconds squared downwards.
What we do to solve a problem like this is, since it isn't 2 dimensions to make it simpler.
That is how we solved this problem.
It's just we can't do a 2 dimensional problem, so we just make it into two 1 dimensional problem.
So we have this arrow, this red arrow pointing off at an angle,
which means that it's hard to tell how much of it is up, and how much of it is sideways.
So the very first thing we always do in a problem like this
is to break it up into components just like we saw with the vector before.
So, we consider the horizontal aspect of the motion
and we'll consider the vertical aspect of motion
and we'll keep them completely separate from each other.
Not let them mix at all because we know how to solve 1 dimensional problems.
So first, going through our horizontal equations,
the horizontal variables that we have, we have no acceleration on the horizontal direction.
This is just because left and right we haven't introduce any air resistance
or anything like this, so nothing that's trying to speed up or slow down your object
from the moment it's been launched.
We launched it and then after that there's nothing pushing it or pulling it in the horizontal direction.
The velocity of the object in the horizontal direction,
just using our equations of motion and plugging in a zero for the acceleration,
is just going to be whatever initial velocity it had in the horizontal direction.
This makes sense if we have no acceleration,
whatever velocity in initially had it will continue to have throughout the entire problem.
And again, this is just in the horizontal direction.
The position we can find in the exact same way by going through our equations of motion
and plugging in a zero for the acceleration in the horizontal direction.
Plugging in that zero we see we have the initial position plus the velocity in the x horizontal direction
times the time. In the vertical direction, it's a little more complicated
because we do have an acceleration. We have acceleration of minus g,
or 9.8 meter per second downwards.
So we'll plug this instead into our equations of motion
and get a velocity that is v minus gt, remember to put in that minus sign in there
for the gravitational acceleration, and then we have a position which also depends on
this acceleration in the way given to us by the equations of motion.
So the vertical position is equal to its initial vertical position
plus the initial vertical velocity, times the time,
minus 1/2 the gravitational acceleration times the time squared.