Bo: Hey guys.
Billy: Hey Bo.
Bobby: Hi Bo.
♫ (lyrics) Flipping Physics ♫
Mr. P.: Ladies and gentlepeople, the bell has rung,
therefore class has begun,
therefore you should be seated in your seats
ready and excited to learn the difference
between instantaneous and average.
Billy: Instantaneous!
Bo: Average.
Bobby: They look the same to me.
Mr. P.: Very nice.
We're gonna learn the difference between instantaneous
and average by way of the term "velocity."
So we have instantaneous velocity and average velocity.
Instantaneous velocity is the velocity
at a specific point in time.
While average velocity
is the velocity over a time period.
So, we have instantaneous velocity,
which is a velocity at a specific point in time,
and we have average velocity,
which is a velocity over a time period or a time duration.
Now, we've actually already had
examples of these two different kinds of velocities,
we just haven't identified them as such.
For example, in your uniformly accelerated motion equations
there are two velocities,
velocity final and velocity initial.
Bobby, are velocity final and velocity initial
in your UAM equations,
are those instantaneous or average velocities?
Bobby: Uh, let's see.
Final velocity would be the velocity at the final point,
which is, that's a specific point in time.
Therefore, it must be an instantaneous velocity,
and the same thing is true for the initial velocity.
They're both instantaneous velocities.
Mr. P.: Correct.
The two variables, velocity final and velocity initial,
from our UAM equations are instantaneous velocities
because they're velocities at specific points in time.
That final point, and that initial point.
Bo, what is the equation we know for velocity?
Bo: Velocity equals displacement over time.
Billy: Actually, it's over change in time.
Bo: Yeah, um, change in time.
And because it's change in time,
that is a time duration, a time period,
so velocity equals displacement over change in time.
That's an equation for average velocity.
Mr. P.: Great.
So you can see we've actually already been working
with instantaneous and average velocities.
Now we're gonna take a look at a graph
and talk more about them using the graph.
We have our position as a function of time graph
Position in meters, time in seconds.
We have on our position axis 2.0 meters and 7.0 meters.
On our time axis we have 5.0, 10.0 and 17.0 seconds.
The blue line indicates the motion of the object.
Geneve, could you please be our object this time?
Could you please walk the graph for us
so we can see what it looks like?
Geneve: Sure, Dad!
Start.
Stop.
Mr. P.: Thanks Geneve.
Now I'm gonna ask several questions.
We're gonna figure out some stuff about this graph.
which will help us to understand the difference
between instantaneous and average velocity.
First, what is the velocity from zero to five seconds?
Now, Billy!
When I ask for the velocity from zero to five seconds,
am I asking for an instantaneous velocity,
or an average velocity?
Billy: Well, because you're asking it
from zero to five seconds it must be an average velocity,
because "from zero to five seconds," that's a time period.
Mr. P.: Yes!
Therefore we need to use the equation for average velocity,
which is change in position over change in time.
And we know that delta always means final minus initial.
Therefore on the top we get position final
minus position initial.
On the bottom, we end up with time final minus time initial.
And then, we know that our final position is five seconds,
and our initial position was zero seconds.
Therefore it works out to be the position at five seconds
minus the position at zero seconds,
divided by the time at five seconds
minus the time at zero seconds.
Bo, could you please look at the graph
and fill in the numbers for me?
Bo: Uh, let's see.
According to the graph,
the position at five seconds is two meters,
and the position at zero seconds is also two meters,
and the time at five seconds is five seconds,
and the time at zero seconds is zero seconds,
so we get 0/5 which is just zero,
so the average velocity from zero to five seconds is zero.
Mr. P.: Yes, the average velocity
from zero to five seconds works out to be zero.
This is how you would solve it mathematically.
But the truth is, you should be able to look at the graph
and know that the average velocity from zero
to five seconds is zero.
Who can tell me how?
(Billy and Bobby speak over each other)
Bobby: Go ahead, Billy.
Billy: Thanks, Bob.
Because if you watch the video or look at the graph,
you can see that Geneve doesn't move
for the first five seconds,
and when someone isn't moving,
their velocity is zero.
Mr. P.: Yes, Billy, that's absolutely correct.
There are other ways of looking at it, though.
So Bobby, what were you going to say?
Bobby: The line is horizontal from zero to five seconds.
The slope of a horizontal line is zero,
and the slope of a position versus time graph is velocity.
Therefore, her average velocity from zero to five seconds
must be zero.
Mr. P.: Great!
Both answers are correct.
Let's move on,
and let's look at the velocity from five to 10 seconds.
So Billy, could you please figure out
the velocity from five to 10 seconds,
and while you do so, could you please tell me
whether you're figuring it out
an instantaneous or average velocity.
Billy: Well, again, five to 10 seconds is a time duration
so you're asking for an average velocity.
So we use the equation velocity equals change
in position over change in time.
We know both of those are final minus initial.
The final is at 10 seconds
and the initial is at five seconds,
so we get the position at 10 seconds
minus the position at five seconds,
over the time at 10 seconds minus the time at five seconds.
Reading the numbers off the graph,
that works out to be...
the position at 10 seconds is seven meters,
minus the position at five seconds,
which is two meters,
divided by 10 seconds minus five seconds,
so we get five over five, which is one,
and with two sig figs we get 1.0 meters per second.
The average velocity from five to 10 seconds
is 1.0 meters per second.
Mr. P.: Very nice.
So far, we've figured out the average velocity
from zero to five seconds is zero
and we figured out the average velocity
from five to ten seconds is 1.0 meters per second.
Bobby, what is the velocity at seven seconds?
Bobby: That would be 1.0 meters per second.
Mr. P.: And how do we know?
Billy: Because the slope of a position versus time graph,
again, is velocity, and the slope of the line
at seven seconds is 1.0 meters per second.
Mr. P.: And Bo, is the velocity of 1.0 meters per second
at seven seconds,
is that an instantaneous or an average velocity?
Bo: Instantaneous velocity,
because it's at the specific time of seven seconds.
Mr. P.: Good.
Bobby, what is the instantaneous velocity
at nine seconds?
Bobby: Also 1.0 meters per second.
Mr. P.: Yes.
And Bo, what is the instantaneous velocity
at 6.9723416 seconds?
Bo: Still 1.0 meters per second.
Mr. P: Yes!
So notice that the average velocity from five to 10 seconds
is the same as the instantaneous velocity
of any point between 5 and 10 seconds.
One word of caution, this is not always true.
Let's go through another example.
We're going to figure out the average velocity
from zero to 17 seconds.
Because it's slightly redundant,
I'll go through and do it this time.
The average velocity from zero to 17 seconds.
We're going to use the equation for average velocity,
change in position over change in time.
Delta changing is always final minus initial.
Final is 17 seconds, initial is zero seconds.
Therefore we have the position at 17 seconds
minus the position at zero seconds,
divided by the time at 17 seconds
minus the time at zero seconds.
That works out to be seven.
The final position minus two the initial position,
divided by 17 the final time,
minus a zero at the initial time.
That works out to be 5/17ths, or 0.29412 with two sig figs
0.29 meters per second,
the average velocity from zero to 17 seconds.
Billy, how can we illustrate
this average velocity on the graph?
Billy: I don't know, how can you show the average velocity
for the whole event on the graph?
Oh, that would mean the slope of a line
that goes from zero to 17 seconds!
Mr. P: Yes!
The slope of this dotted red line
is 0.29 meters per second,
and it represents the average velocity
from zero to 17 seconds,
or the average velocity for the whole trip.
Bo, according to our graph,
was Geneve's instantaneous velocity
ever equal to her average velocity for the whole trip?
Bo: It must have been.
Mr. P.: Be careful, Bo!
Listen to the question.
According to our graph, was Geneve's instantaneous velocity
ever equal to her average velocity for the whole trip?
Bo: Oh, I get it.
The answer is, according to our graph,
she only had instantaneous velocities
of either zero or 1.0 meters per second,
and therefore no, her instantaneous velocity
was never 0.29 meters per second.
However, that's not possible.
Mr. P: Yes!
Remember our graph represents ideal data,
which means at any point between zero and five seconds,
and any point between 10 and 17 seconds,
her instantaneous velocity was always equal to zero.
At any point between five and 10 seconds,
her instantaneous velocity was always equal
to 1.0 meters per second, which means
her instantaneous velocity was either zero or one.
It was never equal to 0.29 meters per second,
which was the average velocity for the whole trip,
because this is ideal data.
But Bo pointed out that this is actually not possible,
which is correct.
In order to understand why it's not possible,
I ask this question.
What was Geneve's instantaneous velocity
at exactly five seconds?
Mr. P.: (laughing) You guys are all correct!
Turns out the velocity at five seconds is undefined.
The reason for that is because the slope
at five seconds is undefined,
therefore you're gonna have an undefined
instantaneous velocity at that point.
Which is not possible.
A real object can't have
an undefined instantaneous velocity.
Geneve, in reality, had to accelerate
from zero to one meter per second,
and then she had to decelerate
from one meter per second down to zero.
Therefore, there should be curves at these points
if we're talking about a real object,
and those curves would look like this.
However, these curves mathematically
are pretty difficult to deal with
and that math is really outside of the scope of this class
so in this class, we're really
only gonna deal with ideal data.
It's a good approximation of reality,
but it's not perfect.
Any questions?
Bo: Nope.
Bobby: No thanks.
Billy: Not yet.
Mr. P.: In that case,
let's do a quick review of the stuff we learned today.
Instantaneous velocity is the velocity
of a specific point in time.
For example, velocity final and velocity initial,
in our uniformly accelerated motion equations
are instantaneous velocities.
We also know that average velocity is the velocity
over a time period or a time duration.
The equation therefore is velocity
equals change in position over change in time.
We used a graph to show that when you have a constant slope,
the average velocity and the instantaneous velocity
are going to be the same.
However, when the slope changes,
for example, all the way from the beginning to the end
we had a change in the slope,
therefore the average velocity was not going
to be equal to the instantaneous velocity.
Ladies and gentlepeople,
I hope you enjoy learning with me today.
I enjoyed learning with you.
Voiceover: Lecture notes
are available at FlippingPhysics.com.
Please enjoy lecture notes responsibly.