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## Graphs of projectile motion

# Projectile motion graphs

## Video transcript

- [Instructor] So in
each of these pictures we have a different scenario. We have someone standing at
the edge of a cliff on Earth, and in this first scenario, they are launching a
projectile up into the air. In this one they're just
throwing it straight out. They're not throwing it up or
down but just straight out. And here they're throwing the projectile at an angle downwards. And so what we're going
to do in this video is think about for each of
these initial velocity vectors, what would the acceleration versus time, the velocity versus time, and the position versus
time graphs look like in both the y and the x directions. So I encourage you to pause this video and think about it on your own
or even take out some paper and try to solve it
before I work through it. So let's first think about acceleration in the vertical dimension, acceleration in the y direction. We're assuming we're on
Earth and we're going to ignore air resistance. We can assume we're in some
type of a laboratory vacuum and this person had maybe
an astronaut suit on even though they're on Earth. What would be the acceleration
in the vertical direction? Well the acceleration due to
gravity will be downwards, and it's going to be constant. We're going to assume
constant acceleration. So the acceleration is
going to look like this. And if the magnitude of the acceleration due to gravity is g, we could call this negative g to show that it is a downward acceleration. Once the projectile is let loose, that's the way it's
going to be accelerated. Now what about in the x direction? Well if we assume no air resistance, then there's not going
to be any acceleration or deceleration in the x direction. So it's just going to be, it's just going to stay right at zero and it's not going to change. And what I've just drawn
here is going to be true for all three of these scenarios because the direction
with which you throw it, that doesn't somehow affect
the acceleration due to gravity once the ball is actually
out of your hands. So now let's think about velocity. So what is going to be the
velocity in the y direction for this first scenario? Well we could take our
initial velocity vector that has this velocity at an angle and break it up into
its y and x components. So this would be its y component. We just take the top part of
this vector right over here, the head of it, and go to the left, and so that would be the
magnitude of its y component, and then this would be the
magnitude of its x component. So the y component, it starts positive, so it's like that, but remember our acceleration
is a constant negative. So our velocity is going to
decrease at a constant rate. So our velocity in this first scenario is going to look something, is going to look something like that. Now what about the velocity
in the x direction? We see that it starts positive, so it's going to start positive, and if we're in a world
with no air resistance, well then it's just
going to stay positive. Notice we have zero acceleration, so our velocity is just
going to stay positive. One of the things to really keep in mind when we start doing
two-dimensional projectile motion like we're doing right over here is once you break down your
vectors into x and y components, you can treat them
completely independently. That something will
decelerate in the y direction, but it doesn't mean that
it's going to decelerate in the x direction. Now what would the velocities look like for this blue scenario? Well our velocity in our y direction, we start off with no
velocity in our y direction so it's going to be right over here. But then we are going to
be accelerated downward, so our velocity is going
to get more and more and more negative as time passes. And notice the slope on
these two lines are the same because the rate of
acceleration is the same, even though you had a
different starting point. Now what about the velocity
in the x direction here? It looks like this x initial velocity is a little bit more than this one, so maybe it's a little bit higher, but it stays constant once again. Now let's look at this third scenario. In this third scenario, what is our y velocity,
our initial y velocity? Well it would look something like that. And our initial x velocity
would look something like that. If we were to break things
down into their components. So our y velocity is starting negative, is starting negative, and then it's just going to
get more and more negative once the individual lets go of the ball. Because you have that
constant acceleration, that negative acceleration, so it's gonna look something like that. And what about in the x direction? Well looks like in the x
direction right over here is very similar to that one, so it might look something like this. I'll draw it slightly higher
just so you can see it, but once again the velocity
x direction stays the same because in all three scenarios, you have zero acceleration
in the x direction. Now last but not least
let's think about position. So they all start in the exact same place at both the x and y dimension, but as we see, they all have
different initial velocities, at least in the y dimension. So let's start with
the salmon colored one. So the salmon colored one, it starts off with a some
type of positive y position, maybe based on the height of
where the individual's hand is. And then what's going to happen? Well it's going to have
positive but decreasing velocity up until this point. At this point its velocity is zero. So its position is going to go up but at ever decreasing
rates until you get right to that point right over there, and then we see the velocity
starts becoming more and more and more and more negative. So it would look something like, something like that. Now what would be the x position of this first scenario? Well if we make this position
right over here zero, then we would start our x
position would start over here, and since we have a constant
positive x velocity, our x position would just
increase at a constant rate. It would do something like that. Now what about this blue scenario? Well this blue scenario, we are starting in the exact same place as in our pink scenario, and then our initial y velocity is zero, and then it just gets
more and more and more and more negative. So it would look something, it would look something like this. Now what about the x position? Well our x position, we had a slightly higher velocity, at least the way that I drew it over here, so we our x position would
increase at a constant rate and it would be a slightly
higher constant rate. So it would have a slightly higher slope than we saw for the pink one. Now the yellow scenario, once again we're starting
in the exact same place, and here we're already starting
with a negative velocity and it's only gonna get more
and more and more negative. So it's just gonna do something like this. It's gonna get more and
more and more negative. It's a little bit hard to see, but it would do something like that. And if the in the x direction, our velocity is roughly the
same as the blue scenario, then our x position over
time for the yellow one is gonna look pretty pretty similar. So this is just a way to visualize how things would behave
in terms of position, velocity, and acceleration
in the y and x directions and to appreciate, one, how to draw and visualize these graphs and conceptualize them, but also to appreciate that you can treat, once you break your initial
velocity vectors down, you can treat the different dimensions, the x and the y dimensions, independently.

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