In problem solving as in street-fighting: Rules are for fools!
(Laughter)
(Applause)
Let's see how far we can go by bending rules
as we estimate the fuel efficiency,
the miles per gallon of a 747.
The fuel is used to fight drag,
the force of air resistance,
what you would feel
if you stuck your hand out of a moving car --
don't try this at home --
or try to run in a swimming pool.
There are at least two ways that you can use
to figure out the drag.
You could spend 10 years learning physics
and you write down the Navier–Stokes equations:
the differential equations of fluid dynamics.
And then you spend another 10 years learning mathematics
to solve for the pressure.
And whereupon you find
that actually there's no exact solution
for the flow around a 747,
or, in fact, for most of the situations
which you want to know.
Rigor, the rigorous approach,
the exact approach has produced paralysis,
rigor mortis.
(Laughter)
We need a different way.
The street-fighting way,
which starts with a home experiment.
Chair please.
Props please.
(Laughter)
Small cone, big cone. Coffee filters.
They're the same shape,
but this one has one-fourth the area.
This one has four times the area, twice the diameter,
but otherwise the same shape.
When I drop them,
how fast do they fall relative to one another?
Is the big one roughly twice as fast?
Are they comparable in speed?
Or is the small one roughly twice as fast?
Take ten seconds and think.
What do you believe? What does your gut tell you?
And then we'll take a vote.
Check with your neighbor.
(Laughter)
(Crowd murmuring)
OK, let's take a vote.
You don't have to agree with your neighbor.
(Laughter)
That's the beauty of democracy.
So, cheer if you believe that the big cone
will fall roughly twice as fast as the small cone.
(Faint cheering)
OK, I hear a few.
Cheer if you believe that they'll be roughly comparable.
(Louder cheer)
And cheer if you believe the small cone
will be roughly twice as fast.
(Loudest cheer)
A lot of cheering for that one.
OK, well, as Feynman said and believed,
in science we have a supreme court: experiment.
So, let's do the experiment!
One, two, three.
(Cheering) (Applause)
They're almost the same.
Within experimental error.
So what does that mean?
What can we use that experiment to tell us?
Well,
the cones fell at the same speed.
They fall in the same air. It has the same density.
The same properties. The same viscosity.
The only things different between the two cones
is this one has four times the area,
the cross sectional area of this one,
and their drag force is different.
How different?
Well, the drag force is equal to the weight.
Because they were falling at a steady speed with no acceleration.
So the drag and the weight cancel.
So we have a very sensitive measure
of the drag force without any force sensors.
All we do is measure the weight.
So this one has four times as much paper as this one.
So it's four times heavier, four times the drag.
Only change, four times the area.
The conclusion: drag is proportional to area.
Not square root of area, not the square of the area.
but just the area.
That's the result of our home experiment
without the rigorous rigor mortis method.
How can we use that?
Well, that one constraint,
along with the next street-fighting tool
of dimensional analysis, solves the drag force.
We match their dimensions.
We match the dimensions of force, drag force on one side
with what we have on the other,
which is area, density, speed and viscosity.
But we already know how to put in the area, just one of them.
That gives us length squared, meters squared.
Now we look and we say, "Oh, there's kilograms over here,
we have to get a kilogram over here."
The only place to get it from is density.
Speed and viscosity, the kinematic viscosity, have no mass in them.
So we put in one density.
Now what we need still is meter squared / second squared,
out of speed and viscosity.
The only way to make it is speed squared.
So there is our drag force.
One experiment for a constraint.
Dimensional analysis for the rest of the constraints.
Drag Force = Area x Density x Speed squared.
How can we use this?
Well, the fuel consumption is proportional to the drag force.
So, let's compare the fuel consumption of a plane with a car.
Rather than calculating the plane from scratch, compare it to a car.
Another street-fighting technique.
So there're three factors in the comparison, in the ratio:
the area, the air density and the speed squared.
Do them one at a time.
So, the area. Well, in the old days of plane travel,
you could lie down on three seats
and there were three sets of those seats.
So three people wide. Plane is about three people high.
So it's nine square people.
A car: Well, from nocturnal activities in cars
you know you can sort of lie down in cars a bit uncomfortably.
(Laughter)
And you can stand up. So it's one square person.
So it's roughly a ratio of ten,
maybe nine or ten.
So the plane is 10 times less fuel efficient for that.
What about air density?
Well, the planes fly high, about Mt. Everest.
So the density is about one third.
So that helps the plane.
But they fly about ten times faster,
600 miles an hour versus 60.
That means planes pay a factor of a hundred, 10 squared.
The result is planes are 300 times
less fuel efficient than cars.
Oh, no. By flying here, did I damage the environment
300 times compared to driving? (Gasp)
What saves it?
300 people on my plane!
So the conclusion is planes and cars
are roughly equally fuel efficient.
(Laughter)
All from that.
(Applause)
So let's say the plane is 30 miles per gallon.
Crossing the country back and forth 6,000 miles,
30 miles per gallon, 2 dollars a gallon.
That's 400 dollars of gasoline.
That's not that different than the price of my plane ticket,
which may explain why airline companies teeter on bankruptcy
and why they charge us for peanuts.
(Laughter)
So connection between the 747 and the cones.
They increase our enjoyment of the world
and expand our perception.
And that, making connections here
was enabled by street-fighting reasoning,
by getting away from rigor mortis.
Making connections is so important
because it builds ideas and isolated facts
into a coherent story.
Imagine each dot is an idea
and the lines are the connections between them.
As I increase the fraction of connections
from 40% to 50%, to 60%,
the big story, the red connection network,
grows to fill the whole space.
That's the long lasting learning.
That's what we want to build in our thinking
and in our teaching.
The goal of teaching should be
to implant a way of thinking that enables a student
to learn in one year what the teacher learned in two years.
Only in that way can we continue to advance
from one generation to the next.
In fifty years, all education
will, I believe and dream, be based on this principle.
Richard Feynman, I think, would have agreed.
Thank you.
(Applause)