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 seat, ready and excited to do
an example problem about average velocity.
Billy: Yes!
Bobby: Yup.
Bo: Sure.
Mr. P.: Let's start by just reading the problem
and then we'll translate the words into physics.
Billy, could you just read
the whole problem first, please?
Billy: Buster and Mr. P. embark on a southward journey.
First, they walk South at 6.5 kilometers per hour
for 1.1 hours.
Then, they stop to take a nap.
For 18 minutes.
Then continue South at 5.5 kilometers per hour
for 1.2 hours.
Part A: what was their average velocity
for the whole trip?
Part B: what was their displacement
for the whole trip?
Mr. P.: Great, Billy could you, again,
just start reading the problem and Bo,
could you please translate?
Billy: Buster and Mr. P. embark
on a southward journey.
First, they walk South at 6.5 kilometers per hour
for 1.1 hours-
Bo: Please stop.
Write, all South because everything's South
in this problem.
Velocity equals 6.5 kilometers per hour
and the change in time, or delta T,
equals 1.1 hours.
Mr. P.: Yes, Billy, please keep reading.
Billy: Then, they stopped to take a nap
for 18 minutes and then continue South
at 5.5 kilometers per hour for 1.2 hours.
Part-
Bo: Stop.
18 minutes is change in time.
So is 1.2 hours.
So another change in time.
So 2 more changes in time.
Then 5.5 kilometers per hour is velocity.
Bobby: How can the change in time be 1.1 hours,
18 minutes, and 1.2 hours?
Mr. P.: That is a great question Bobby.
It turns out that that can't be true.
Bo: Can we split the problem into 3 different sections?
Mr. P.: Yes, we can split this problem
into different sections, or different parts.
Yeah, so there are 3 different sections.
You could see we have the change in time
for part 1 is 1.1 hours, the change in time
for part 2 is 18 minutes, and the change in time
for part 3 is 1.2 hours.
We're going to take subscripts on these
2 velocities, Bobby, can you please help me?
Bobby: 6.5 kilometers per hour is the first velocity
and 5.5 kilometers per hour is the second velocity.
Billy: There are 3 parts, so I think
there should be a third velocity.
Bo: I missed 1.
The second velocity is actually 0
because they stopped to take a nap.
Then the third velocity is 5.5 kilometers per hour.
Mr. P.: Very good.
People often, for some reason, overlook 0's
because they think they're not important.
However, 0's are very important and it's
very important that you write those down.
So Buster and I stopped for 18 minutes, part 2
and we took a nap.
So we weren't actually moving, so our average
velocity for part 2 was equal to 0.
Billy, could you please keep reading?
Billy: Part A: what was their average
velocity for the whole trip?
Part B: what was their displacement for
the whole trip?
Bo: I don't know what to tell you to write.
Mr. P.: Well, I don't know what we're looking for.
Bo: Oh, right.
Velocity equals question mark and delta X
equals question mark because those are the 2
things we're trying to find.
Mr. P.: You'll notice there are subscripts
on all of our knowns, but what we're trying to find,
we don't have any subscripts on.
We do need subscripts on those.
Billy, what subscripts should we have?
Billy: Those are both for the total trip,
or the whole trip, so we need to illustrate
that they're both for the total trip.
So, do we write total?
Mr. P.: Absolutely.
We put a subscript of T on both of them.
T to illustrate that we're looking for
the velocity and the displacement for
the total trip, the whole thing.
So T for total.
Class, have we actually started solving
this problem yet?
Billy: Uh, no.
Bobby: Nope.
Bo: Not yet.
Mr. P.: I have found in my experience,
that students have a tendency to just rush
through a problem and try to solve the problem
even before they understand it.
It is very important that you slow down,
write down everything that you know,
what you're trying to find, and understand
the problem before you try to solve it, please.
Bobby: Should we convert the 18 minutes to hours, now?
Mr. P.: Bobby, that's a great idea.
If you look at the givens we have kilometers
as our linear dimension and as our time dimension,
we have hours, hours, hours, hours, and this
is the only one that's not in hours,
it's in minutes.
So, yeah Bobby, could you convert 18 minutes
to hours, please?
Bobby: There are 60 minutes in an hour
and we need to cancel out minutes, so multiply
by 1 hour over 60 minutes.
Bo: That works out to be exactly .3 hours.
Mr. P.: So we multiply the 18 minutes,
the change in time from part 2, times 1 hour
over 60 minutes.
This means that the minutes are on top
and bottom, therefore they cancel out.
We're left with only hours, or 18 over 60,
or 0.3 hours.
That's the change in time for part 2.
Now, when we have a problem that has multiple parts
sometimes I find that students see it better this way.
Again, I find that sometimes I find that students
understand the givens better when there
are multiple parts when we set it up
in a table form.
For example, we have in this column the part,
in this column the change in time, and this column
the velocities.
You can see, I put the dimensions at the top
so that we don't have to put them
throughout the table.
So, for example, for part 1 the change in time
is 1.1 hours and the velocity for part 1 is
6.5 kilometers per hour.
You can see all the information in the table, there.
Bo, why don't you now work on part A,
finding the average total velocity for,
again, the total trip?
Bo: Well, I am going to learn from last time
and start with an equation.
Mr. P.: Thank you Bo.
The equation is always a good place to start.
Bo: You're welcome.
We know velocity equals change in position
over change in time.
So we have each velocity and change in time.
So we can just solve for the displacement
for each part.
Mr. P.: Great.
A good thing to do before we plug in numbers
and solve for the displacement is actually
to rearrange the equation and solve for
that displacement, just in terms of variables,
rather than plugging in numbers first.
Because we have multiple parts and we would
just do it several times over and over again.
So Billy, could you please rearrange how to
solve this equation for the change in position,
or the displacement, please?
Billy: Multiply both sides by delta T.
Mr. P.: So, you can see if we multiply both sides
by the change in time, the change in time
cancels out on the right hand side
and we're just left with the displacement.
So the displacement ends up being equal
to the change in time multiplied by the velocity.
At this point, we can now go through and plug in
those subscripts for each individual part.
So let's go through and do that.
So you can see we can substitute in the numbers
for part 1.
We have the displacement for part 1 is equal
to the change in time for part 1 multiplied
by the velocity for part 1.
So the change in time in part 1 was 1.1 hours
and the velocity from part 1 was 6.5 kilometers
per hour.
The hours cancel out and we're just left
with kilometers.
Who can tell me how many kilometers do we have
as a displacement for part 1, please?
Bobby: That works out to be 7.15 kilometers.
Mr. P.: So our displacement for part 1
is 7.15 kilometers.
Oh, you can see, again, I'm not writing South
on these displacements or velocities
because everything is in the southern direction.
We can do the same thing for part 2 now.
The displacement for part 2 is equal to
the change in time from part 2 multiplied
by the velocity from part 2, which is going to be
0.3 hours times 0 and anything times 0
is going to be equal to 0.
Hopefully, you should have recognized early on,
that the displacement for part 2 is going
to be equal to 0 because we're taking a nap.
We're clearly not moving.
But this is what it would look like mathematically
and that's important to be able to show.
Again, we can do the same thing for part 3.
So again, for part 3 we have the displacement
for part 3 is equal to the change in time
for part 3, multiplied by the velocity for
part 3, which is going to be equal to 1.2
hours times 5.5 kilometers per hour.
Again, the hours cancel out and, again,
we get kilometers for our dimensions.
What do we get for our displacement
for part 3, please?
Billy: Delta X equals 6.6 kilometers.
Mr. P.: Oh, hey.
Now that we have each individual displacement,
we can solve part B.
Bobby: Yeah, you just add all the displacements
together to get the total displacement.
Mr. P.: Yeah, that's true.
Let's walk through that.
The total displacement is going to be
equal to the displacement for part 1,
plus the displacement for part 2, plus
the displacement for part 3.
All we need now is the numbers.
What do we get now, for our total displacement?
Bobby: 13.75 kilometers.
Billy: Sig figs. Bo: Rounding.
Bobby: Oh, um 14 kilometers South.
Mr. P.: So, it turns out, that we get our answer
for part B actually before we get the answer
for part A.
The total displacement is equal
to 14 kilometers South.
14 kilometers in 2 sig figs.
That would be the magnitude of the displacement
and South being the direction of the displacement.
Back to the problem, we need to solve part A.
Bo?
Bo: Oh, let's see.
We need the total average velocity.
So let's go back to the original equation
for velocity and we add subscripts for total.
Mr. P.: Right.
So we just come back to the original equation
for velocity, velocity equals change in position
over change in time.
We just add T's, a set of subscripts, for total.
So the total velocity equals the displacement
total divided by the change in time total.
In part B we actually already figured out
the displacement total, so all we need
at this point, is the change in time total.
The time for the entire event.
Let's walk through that.
So now, what do we get for our total change in time?
Billy: 2.6 hours.
Mr. P.: Now that we have our change in time
total and our displacement total we can come back
to the equation for velocity total
and we can just substitute in the numbers.
So, what do we get for our total velocity, please?
Bobby: 5.28846 and that rounds to 5.3
kilometers per hour South.
Mr. P.: Yes.
Notice, with sig figs.
5.3, so we have 2 sig figs.
So 5.3 kilometers per hour's the magnitude
of the velocity and South being the direction.
So, the total average velocity is 5.3
kilometers per hour, South.
Billy: Mr. P. why can't we just add
the 3 velocities together and divide
by 3 to get our average velocity?
Mr. P.: That is a great question.
It's one that I often get asked at this point.
Let's walk through the numbers there,
and we can talk about why this doesn't work.
A lot of people simply want to add
the 3 velocities together and divide by 3
in order to get the average velocity.
You can see, when we do that, we get 4.583
repeating, which is clearly not equal
to 5.28846.
The question is, "Why doesn't this work?"
Under what circumstances would this work?
When could we simply add the 3 velocities
together and divide by 3 to get the average velocity?
Bobby: I think that only works if each part
lasts for the same amount of time, right?
Mr. P.: Yes Bobby.
The only time that this would work,
when we could add the 3 velocities together
and divide by 3 to get the average velocity
is if we drive at those velocities
for the exact same amount of time.
Clearly, that's rarely going to be the case.
So you're going to have to go through
and solve it the way that we talked about
right here.
That is my velocity example involving multiple parts.
I hope you enjoyed learning with me today.
I enjoyed learning with you.
Voiceover: Lecture notes are available
at flippingphysics.com.
Please enjoy lecture notes responsibly.
Mr. P.: Do I look good?
Therefore you should be seated in your seat.
That's tough to say.
And this is going to be in velocities ...
Ah, blah, blah, blah [snaps fingers].
Try again.
Uh ...
This is, yeah.
So, again, I'll find that some styles, um ...
Would be if we're moving at the, blah.
Ah.
Is if we're driving at that average,
uh, at, uh [frustrated noise].