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There's more than one way to measure how things move.
We can use equations,
or you can represent movement by drawing it on a graph.
Graphs are an amazing way to show a lot of information
about movement, easily and accurately.
Let's look at how to draw graphs of motion.
It can actually be pretty cool.
Say we want to graph how Summer got from a food truck
to her car, carrying a sandwich.
What direction did she go, how fast, how far,
and how long did it take her to get there.
Or, in physics terms, what is the displacement,
velocity, acceleration, and time?
You can do all of those kinematic equations,
or you can draw it on a graph.
Now graphs don't sound so bad, do they?
We wanna graph Summer's position
at a certain moment in time.
Let's say five seconds,
which happens to be when her phone rings.
We'll do it on a position versus time graph.
The vertical axis shows her position in meters.
The horizontal axis shows time in seconds.
First, we find five seconds on the time axis.
Then we draw a tangent line,
which is a straight line that touches the graph
and then runs infinitely far from that point
in both directions.
To find the instantaneous velocity,
we need to find the slope of this line.
We pick two points on the tangent line,
and label the first point's coordinates x1 and y1.
And the second point's coordinates x2 and y2.
The slope is represented by a change in position
found on the y-axis, delta y,
divided by the change in time found on the x-axis, delta x.
Usually the slope will have units
where the y units are on top
and the x units are on the bottom.
Plug the two points into the slope equation,
which, in this case, can be written as
final position minus initial position,
divided by final time minus initial time.
And that's the graph of Summer's instantaneous velocity
at five seconds, which is 0.5 meters per second.
Now let's graph Summer's average velocity
for her entire journey.
Her average velocity is her total displacement
divided by the time elapsed.
So take the two points,
one at the beginning of her journey
and one at the end.
Her average velocity is the slope of the line
connecting these two points.
Find the slope of the line connecting the two dots
with the same slope equation.
The change in y, or position,
divided by the change in x, or time.
You can pick any two points on the line when you do this.
So, for example, you can pick the point (0,0)
and (24,4).
So, our final position minus our initial position,
divided by our final time minus initial time,
is your slope.
In the above points, y2 is 4, y1 is 0,
x2 is 24, and x1 is 0.
When you plug these numbers into the equation,
you get four minus 0 divided by 24 minus 0,
which equals a slope of 0.17.
The y-axis is the position in meters,
and the x-axis is the time in seconds,
so the units are meters per second.
A straight line going up
on a position versus time graph
represents a positive constant velocity.
So we are moving away from the origin
in the positive direction.
Let's make the positive direction
going away from the food truck.
A horizontal line means Summer has stopped moving.
A line going down indicates she's moving
at a constant velocity back toward the origin
in a negative direction.
Let's say Summer forgot to get her change
and has to head back to the truck.
A line that curves up indicates an object is speeding up.
Summer is late, so she turns around
and starts to hustle and forgets about getting the change.
And the line that curves down indicates slowing down,
or negative acceleration.
Summer is so late, there's no point in hurrying.
So that's how we graph velocity and acceleration
on a position versus time graph.
Now let's find Summer's displacement.
Between time 6 seconds and 20 seconds.
You do that by finding her change in position
between two points, or delta x.
Another way to say it is x final minus x initial.
So at time 6 seconds,
Summer's position was 3 meters,
and at time 20 seconds,
Summer's position was 3 meters.
Her displacement would be 3 meters minus 3 meters,
or 0 meters between time 6 and 20 seconds.
So that's motion on a position versus time graph.
Now let's look at motion using a velocity versus time graph.
In this kind of graph, velocity is on the y-axis,
measured in meters per second,
and time is on the x-axis, measured in seconds.
Let's take the data from the position versus time graph
and plot it on the velocity versus time graph,
and we'll see how her velocity has changed over time.
At A, Summer was moving at a constant velocity,
so let's graph that first.
Between time 0 and 7 seconds,
we draw a horizontal line showing Summer's velocity.
Now let's graph B, where she was stopped.
So, what is her velocity?
You can probably guess.
It's 0 meters per second
between time 7 and 19 seconds.
We graph that as a horizontal line
at 0 meters per second.
What about when Summer started going in the negative direction
at a constant velocity back to the food truck?
On a velocity versus time graph,
that would be a horizontal line with a negative y value.
Summer then turned back around in the positive direction
and started to speed up.
This is a line that has a positive slope
up and to the right.
Summer then slowed down in the positive direction,
so that's a line that goes down to the right.
You can graph instantaneous and average acceleration
on a velocity versus time graph,
just like we graph instantaneous and average velocity
on a position versus time graph.
The method is exactly the same.
For instantaneous acceleration,
draw a line tangent to that moment in time
on the graph line, and find the slope of it.
For average acceleration,
draw a line connecting your time interval,
and find the slope of it.
There's one more kind of graph that we haven't shown you,
and that's acceleration versus time.
Let's take a look at an example of that
using Summer and her sandwich.
Between A and C, she is not accelerating.
She was either at a constant velocity or stopped.
So, her acceleration was 0.
At D, she sped up in the positive direction
because she was late.
And then at E, she slowed down in the positive direction
because she was too late to worry about it.
So that's why it's on the negative part of the y-axis.
So we use Summer on her lunch break
to show you how to graph motion
using a position versus time graph,
a velocity versus time graph,
and an acceleration versus time graph.
That's it for this segment of "Physics in Motion".
We'll see you next time.
For more practice problems, lab activities,
and note-taking guides,
check out the "Physics in Motion" toolkit.