ANDREW DUFFY: So our goals for this session are the following.
We'll see how speed and velocity are different,
learn how to calculate average values, and we'll
learn how to find instantaneous values of speed and velocity
and we'll often do that through a graph.
So what is speed and how does it compare to velocity?
Speed is simply a scalar representing how fast something is moving.
Velocity combines speed with the direction of motion.
We can also define velocity as the rate of change of position,
the time rate of change of position.
The speed at any instant is the magnitude
of the velocity at that instant.
Here's an example of speed, 50 kilometers per hour.
Convert that to a velocity.
You have to combine that with the direction, 50 kilometers per hour
north, for instance.
Sometimes, we want to know the average values averaged over time
of either the speed or the velocity.
How do we do that?
Well, average speed is some quantity over total time.
What do you think it is?
Average velocity is some different quantity over the total time.
What do you think that is?
So average speed turns out to be the total distance traveled
divided by the total time, where average velocity is the net displacement
over the total time.
Well, look at how these are different.
Sometimes, they are.
Sometimes, they aren't.
They are different if there's a change in direction
and of course, actually, they're different all the time
because even if the magnitudes are the same,
the average velocity has a direction associated
with it, while the speed does not.
So let's say you walk 10 meters to the right and then four meters
left and another 8 meters to the right and all this
happens at a total time of 10 seconds.
What's your average speed?
So the average speed, you just say, well,
what's the total distance that I traveled?
Add up all the individual distances, 10 plus 4 plus 8, 22 meters.
Divide by the time.
That's your average speed, 2.2 meters per second.
Average velocity, on the other hand, let's see how you calculate that.
So now, you have to worry about the different directions.
So here, we'll say-- it's a one-dimensional situation.
We'll say positive's to the right.
So we do plus 10 meters minus 4 meters for the one left
and then 8 meters to the right plus 8 meters.
Add that up.
You get a net displacement of 14 meters.
So you end up 14 meters to the right of where you started.
Average velocity is 14 meters divided by the plus 14 meters,
in fact, divided by the 10 seconds, plus 1.4 meters per second.
Here, the plus indicates that the average velocity
is directed to the right.
So let's now think about instantaneous values.
Sometimes, we want to know right at this very instant how fast are you going.
What direction are you traveling in?
So if you're driving in your car, what would
use to find your instantaneous speed?
And of course, that's the job of the speedometer.
Let's say you're driving from Boston to New York City.
What would you use to find your average speed?
And so you say, well, I can get my total distance covered from the odometer
and I've got a clock in my car or on my wrist.
That'll tell me the total time.
Divide distance by time.
That's your average speed.
And along this drive, you're driving on the Mass Pike
in Massachusetts, the turnpike.
You pass the state trooper.
The trooper is interested in what?
Your average speed?
Your instantaneous speed?
And of course, it's your instantaneous speed.
And what they get in their radar gun should
match what's on your speedometer.
So you get no excuse if you're speeding.
So let's think about instantaneous velocity.
So here's our equation.
Average velocity is net displacement over total time
or we can write it in a much more compact form, v
with a bar on top-- a little bar on top represents average of a quantity--
is delta x-- the x has a vector symbol on top-- divided by delta t.
The instantaneous velocity happens to be the limit of delta x over delta t
as delta t approaches 0.
That's sort of a calculusy way to find it.
And the instantaneous speed is the magnitude
of the instantaneous velocity.
However, we can often get these instantaneous values from graphs.
So what do you think is the best way to complete these sentences?
Fill in the blanks.
The instantaneous velocity is the slope at a particular instant
on a position versus time graph.
The displacement is the-- note this is a much
larger blank-- for a particular time interval
on a velocity versus time graph.
Well, that would be the area under the curve.
So let's see how these work.
So we'll look at that statement in particular.
I'm going to focus on a position versus time graph.
So we'll do two examples.
First, we'll do a constant velocity situation.
So we 'll have a truck going at constant velocity.
And you can see that the slope of the position versus time graph
has a constant slope.
That's indicative of the velocity being constant.
Great.
On the other hand, if we have an acceleration instead
and the acceleration, the rate of change of velocity, is constant,
then the slope of the position graph steadily changes, like so.
So here, the truck starts from rest, gradually gets faster and faster
and faster as time goes by, and you can see the slope of the graph
steadily increases.
So that's a big difference on the position
versus time graph for constant velocity or constant acceleration.
What do the velocity versus time graphs look like for those two cases?
So if velocity is constant, we get a nice rectangle
as the area under the curve.
So the velocity here is constant at three meters per second.
The truck does that for 10 seconds.
You can find the displacement as the area under the curve
and it's a rectangle.
So it's easy to find the area of a rectangle.
Do the height, three meters per second, times the width, 10 seconds-- 30 meters
the truck moved.
What happens in the constant acceleration case?
Well, now the velocity graph has a constant slope.
That slope happens to be the acceleration, by the way.
Now, we get the area under the curve.
It's a triangular area.
So we can do 1/2 the base times the height
to find the area under the curve.
So it's 1/2-- the height there is five meters per second of a triangle
and the base is 10 seconds.
So that works out to 25 meters displacement.
So that's an introduction to speed, velocity, average values,
instantaneous values, and also how we can use
graphs to figure out various things about speed and velocity
and displacement, as well.