Bo: Hey, guys.
Billy: Hey, Bo. Bobby: Hi, Bo.
♫ (lyrics) Flipping Physics ♫
Mr. P. : Ladies and gentle people,
the bell has rung
therefore glass has begun
therefore you should be seated in your seat
ready and excited to figure out
the average speed necessary for pole position.
Billy: Yep.
Bobby: Absolutely.
Bo: Let's do it.
Mr. P.: All right, let's get right to the problem.
Bobby, could you please read the problem.
Bobby: During the 2010 Indianapolis 500,
Helio Castro-
Mr. P.: Ah, yes, the name.
I don't know the name either.
Kate, could you please tell us how to pronounce it?
Kate: Helio Castroneves.
Bobby: Okay, let's just start over.
During the 2010 Indianapolis 500,
Helio Castroneves won pole position
by averaging 228.0 miles per hour or mph
for 4 2.500 mile laps.
If he averaged 222.0 mph for the first of 2 laps,
what must his average speed have been
for the last 2 laps?
You may assume the number of laps is exact.
Mr. P.: Good, now let's translate to physics.
Bobby, could you please read,
and Billy could you please translate.
Bobby: During the 2010 Indianapolis 500,
Helio Castroneves won pole position
by averaging 228.0 miles per hour
for 4 2.500 mile laps.
Billy: Please stop.
Speed equals 228.0 miles per hour
and distance equals equals 10 miles.
Mr. P.: Sure. The speed is 228.0 miles per hour,
and the distance is equal to 10 miles,
but notice I didn't write just the distance is
equal to 10 miles.
Instead I wrote down the given values.
It's always important to know
what your knowns are and are known
in this particular case
is that the distance per lap is 2.500 miles;
therefore, the distance traveled
is going to be equal to 4 laps
times 2.5 miles per lap,
and therefore 10 total miles.
Oh, and I use s for speed and d for distance.
Bobby, please keep reading.
Bobby: If he averaged 222.0 miles per hour
for the first 2 laps,
what must his average speed have been
for the last 2 laps?
You may assume the number of laps is exact.
Billy: Uh, well, then speed equals
222.0 miles per hour,
and distance equals 2 laps times 2.5 miles,
or 5 miles.
Bo: We need subscripts because we can't have
the same variable equal to different numbers.
That doesn't make any sense.
Mr. P.: Bo, you are correct.
We do need subscripts,
and Bo, what do you think
those subscripts should be?
Bo: 228 is the first speed so put a subscript
of 1 on it,
and 222 is the second,
so put a subscript of 2 on that one.
Bobby: Actually, I think the 228 miles per hour
is not part of the trip,
but actually for the whole trip, the whole thing.
So you should use t for total
for the speed for 228 miles per hour.
The same is true for the 10 miles.
That distance should be distance sub t
for the total trip.
That means that the 222 miles per hour
is just for the first part so put speed sub 1,
and then the 5 miles that distance
would be distance 1.
Mr. P.: We don't actually have
what we're trying to find on the board yet.
Billy, could you tell me what are we trying to find?
Billy: We are trying to find
the average speed for the second part
so write speed 2 or s sub 2 is equal to "?".
Mr. P.: Good, now we have all the known values
on the board,
and what we're trying to find.
One thing I'm going to do now is
I'm going to draw a visual
that's going to help people understand
what's going on here.
I always love visuals.
All right, I know the track for the Indy 500
is actually an oval; however,
it's much harder to visualize
if you draw it as an oval
so I've just drawn it as a straight line
which works out fine.
You can see we have the total track.
I've split that into Parts 1 and Part 2
for the first 2 laps and the second 2 laps.
You can see that we know
the average speed total and the distance total.
We know the average speed for part 1
and the distance for part 1.
We also know the distance for Part 2,
but the 1 thing we don't know is
the average speed for Part 2
which is what we're trying to find.
Bobby, what would you like to do next?
Bobby: Well, let's start with the equation.
Let's start with the equation for speed.
Speed equals distance over time.
Mr. P.: The equation is always
a great place to start.
In this particular case,
speed equals distance over time,
or s = d over t just using the letters.
You can see that we know a lot of speeds
and we know a lot of distances,
but we don't know any times.
Bo, what do you think we should do next?
Bo: We need to solve for time.
We could multiply both sides by time
to cancel out time on the right-hand side.
That leaves us with time multiplied by speed,
and that's equal to distance.
Then we could divide both sides
by speed to cancel out speed on the left-hand side,
and that leaves us with time is equal
to distance divided by speed.
Mr. P.: Very nice, Bo.
You could see we have speed equals
distance over time.
Our goal is to actually solve for time
as that's the 1 thing we don't really know.
So we multiply both sides by time.
Time cancels out on the right-hand side.
We're left with time times speed is
equal to the distance; therefore,
we can divide both sides by the average speed
and we end up with average speed cancelling out
on the left-hand side.
We end up with the time is equal
to the distance traveled
divided by the average speed.
Now we can look at specifics.
Let's look at Part 1 for example.
So we have all the information we need for Part 1
so we could just put in a 1 for our subscript
so we have time 1 is equal to distance 1
divided by average speed 1.
The distance for Part 1 was 5 miles,
and the average speed for Part 1
was 222 miles per hour.
Therefore, what do we get for our time for Part 1?
Billy: 0.0225225 kilometers per, no, not kilometers,
miles per hour.
No, that's not.
What are the dimensions?
Mr. P.: That is a great question.
We need to figure out the dimensions
for time for Part 1,
I find a lot of students do get confused
by this so let's walk our way through solving
for the dimensions for time for Part 1.
We know time equals distance
divided by the average speed
so we can just substitute
in the dimensions for those.
The distance is in miles
and the average speed is in miles per hour.
In order to solve this,
to figure out what this actually represents,
we need to hearken back a few years
to a phrase that you probably remember
which is "flip the guy, and multiply."
Bobby: That's right.
Flip the guy, and multiply.
Billy: Oh, yeah.
Bo: I don't think I've ever heard of that.
Mr. P.: The guy that we need to flip and multiply
is miles per hour; therefore,
we flip him.
We get hours over miles
and we multiply by what's on top.
Now what we have on top is actually miles over 1
which means we end up with miles over 1
times hours over miles.
The miles cancel out and we end up with hours.
We get 0.0225225 hours for a time for Part 1.
We can now do the same thing for the total trip.
You can see for the total trip
we're going to get the time total
is equal to distance total divided
by the average speed total.
The distance total was 10 miles,
and the average speed total was 228 miles per hour.
Therefore, the time for the total trip is ...
Bobby: 0.0438596 hours.
Mr. P.: Great. Now that we figured out the time
for Part 1 and the time total,
I want to make sure we understand
what these numbers actually represent.
Bo, please tell me what do these 2 numbers
actually represent?
Bo: Well, 0.0225225 hours is the time
that the guy spent driving at 222 miles per hour,
and 0.0438596 hours is the time
that the guy spent driving at 228 miles per hour.
Mr. P.: Bo, that's pretty close.
I do want to elaborate on it though.
This time, the time for Part 1, 0.0225225 hours,
this is the time where he
averaged 222 miles per hour.
He didn't actually go that speed that whole time.
We just know that he averaged
that amount of speed during that time.
We used this many hours for Part 1.
In addition to that,
we know that this is the total time
that we can use for the 4 laps or 10 miles.
Again, we know we need to average
228 miles per hour for that time,
but we're not going to travel
at exactly 228 miles per hour.
It's an average time.
This is the total time that we could have possible
if we have more time than this,
and it takes him more time than this
then we're actually going to average less
than 228 miles per hour,
and he's not going to get pole position.
So Bobby, what next?
Bobby: Oh, we can figure out the time
for Part 2 now.
Bo: Yeah, it's just 0.0438596 minus
the 0.0225225.
Mr. P.: Careful.
Please remember always start with an equation.
We know that the total time is equal
to the time for Part 1 plus the time for Part 2.
Therefore, we can subtract the time
for Part 1 from both sides.
We get the time for Part 2 is equal
to the total time minus the time for Part 1.
Therefore, the time for Part 2 is going
to be equal to 0.0438596 hours
minus 0.0225225 hours.
So Time 2 is ...
Billy: 0.0213371 hours.
Mr. P.: Again, just to make sure we fully understand
what this time represents,
Bobby please tell me what does
0.0213371 hours represent?
Bobby: 0.0213371 hours that's the time
for Part 2, or the time left
for the remaining 2 laps
such that Helio Castroneves
can average 228 miles per hour.
Mr. P.: Exactly.
What now, Bo?
Bo: Uh, we can go back to the original equation
for speed, speed equals distance over time
because we know the distance for Part 2
and the time for Part 2
so we can substitute in the information
for Part 2 and solve for the speed for Part 2.
Mr. P.: You can see now that we have
the time for Part 2,
we can figure out the speed for Part 2.
It's just going to be the distance for Part 2
divided by the time for Part 2.
So the average speed for Part 2
is just going to be 5 miles divided
by 0.0213371 hours.
Now, before we solve for this number,
I want to take a moment
and just think about the way students often
look at this problem.
Going back to the very beginning,
students often look at this and say,
"Well, we need to average 228 miles per hour,
"and for the first half we averaged
"222 miles per hour which is 6 miles per hour
"less than 228 miles per hour; therefore,
"for the second half we must average ..."
Bo: Uh, 6 miles per hour more for the second half.
That would be 234 miles per hour,
but something tells me that's not right
because you're asking the question.
Mr. P.: Bo, you are correct that this is incorrect.
This basic idea comes from here.
Regardless of the fact that we went through this
a couple of lessons ago,
students still have this idea
that the average speed is going to be
going the speed for Part 1
plus the speed for Part 2 divide that by 2.
In other words, 228 is equal to 222
plus the speed for Part 2 divided by 2.
Multiply both sides by 2.
We get 2 times 228 is equal to 222 plus
this average speed for Part 2.
Therefore, subtracting 222 miles per hour
from both sides we get the average speed
for Part 2 is equal to 2 times 228 minus 222
which gives you the average speed
for Part 2 would be 234.0 miles per hour.
But again, this is not correct.
What do we actually get
for our average speed for Part 2
by solving this problem correctly?
Billy: 234.333 miles per hour
which rounds with 4 sig figs to 234.3 miles per hour.
And we don't need direction
because this is speed and not velocity.
Mr. P.: Note again, 234.0 is not the same thing
as 243.3 miles per hour.
In other words, this is not correct.
You can't add the 2 speeds together
and divide by 2 to get the average speed.
When is the only time that this actually works
where you can add the 2 speeds together
and divide by 2 to get the average speed?
Please, Bobby, when is that true?
Bobby: Uh, well I think that's only when
the object is moving at those speeds
for the same amount of time.
In this problem it's the same distance
which clearly is very different.
Mr. P.: Great. Please remember you can only
do this when it's for the same amount of time
not the same amount of distance.
Ladies and gentle people, so ends my ...
Bobby: Hang on.
I have a question.
Why did we give our answer
with 4 significant digits?
Shouldn't it have just been 1.
The total distance only had 1 sig fig,
and the same is true for distance
for Parts 1 and 2, 10 miles and 5 miles.
Each of those only have 1 sig fig.
Mr. P.: Okay, that's a good question.
Billy and Bo, what do you guys think?
Bo: Uh.
Billy: Those distances weren't given values.
They weren't knowns.
The distance per lap was the known value
and that had 4 sig figs in it.
The total distance and the distances
for Parts 1 and 2 were derived
from the distance per lap
so we have to go back to the sig fig
for the knowns
not the sig figs for the derived values,
go all the way back to the knowns.
Mr. P.: Yes. Please note that I always
write down our givens.
We need to always do that
with the correct number of sig figs,
but we don't have to carry the correct number
of sig figs throughout the problem.
You simply have the correct number
of sig figs at the very beginning,
and at the very end.
Oh, and speaking of sig figs,
1 thing I do have to confess is
that Helio Castroneves in 2010 in the Indy 500
actually qualified for pole position
with a speed of 227.970 miles per hour.
They actually carry their, they figure out
their average speeds to 6 sig figs
which makes sense.
I cut it down to 4 sig figs
so it would be easier to show
that this is incorrect which is an important piece.
I hope that's okay with you.
Bo: Sure, doesn't bother me.
Billy: Okay, I guess that makes sense.
Mr. P.: All right, that was my example problem
figuring out the average speed necessary
to qualify for pole position.
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.