It was discovered by Galileo
refined by Isaac Newton
and in the hands of Albert Einstein,
provided a theory of the mechanics of the cosmos.
It was one of the deepest mysteries in all of physics.
All bodies fall
with the same constant acceleration.
In a vacuum, all bodies fall
with the same constant acceleration.
That's it.
That's the law of falling bodies.
Doesn't seem like much to get excited about.
And yet, just look at what it says.
First of all, it says that the effect of gravity on all bodies
is the same, regardless of their weight.
From Galileo to Isaac Newton, right down to Einstein,
that's been one of the central mysteries in all of physics.
... says that bodies fall with constant acceleration.
It's almost impossible, even to understand what that means
without a marvelous mathematical device called the Derivative.
We'll see today what that means.
And finally, profound and important to all this is,
it violates our simplest intuition
Because it happens only in a vacuum
not in the world we're familiar with.
For all of us,
the effect of the Earth's gravity was probably our first encounter with the laws of nature.
And whether or not we understand how gravity works
we have an innate fear of what it does.
But exactly what is the effect of gravity?
Some bodies fall to the Earth quickly and directly
while others behave quite differently.
In some cases,
the pull of gravity can be resisted almost indefinitely.
To make any sense at all about how and why bodies fall
we need to separate the effect of gravity on a falling body,
from the opposing effect of the air through which the body is falling.
In other words,
we have to imagine a body falling not through the air
but through a vacuum.
For instance,
what happens if a penny and feather
drop simultaneously from the same height?
They behave exactly as we would expect
each falling at a very different rate than the other.
But that's only because of the effect of air resistance on the two objects.
In a vacuum, a penny, a feather, and any other object,
will fall at exactly the same rate.
With virtually no air remained inside the glass tube,
the penny and feather are now in a vacuum.
When the penny and feather are released,
we'll witness the law of falling bodies in action.
Without the effect of air resistance, in other words in a vacuum,
all bodies, regardless of their weight,
will fall at exactly the same rate.
When Apollo 15 astronaut David Scott explored the airless surface of the Moon,
he couldn't resist repeating this classic experiment for all the world to see.
"Here in my left hand I have a feather,
in my right hand a hammer,
and I guess one of the reasons we got here today
was because of a gentleman named Galileo a long time ago,
who made a rather significant discovery
about falling objects in gravity fields.
And we thought that where would be a better place to confirm his findings
than on the Moon?
And I'll drop the two objects and hopefully,
they'll hit the ground at the same time.
How about that?
Looks like Mr. Galileo was correct in his findings."
Mr. Galileo was correct.
Nearly 400 years ago,
at a time when all the world believed that heavy bodies fall faster than lighter ones,
Galileo realized that in a vacuum,
all bodies should fall at the same rate.
Galileo couldn't produce a vacuum,
but he could imagine one.
He pictured a heavy body attached to a lighter one.
Would this compound body, he asked,
fall faster or slower than the heavy body alone?
If the lighter body did fall more slowly,
it should slow down the heavy body.
So the compound body should fall more slowly than the heavy body alone
But the compound body is actually heavier than the heavy body alone,
therefore, the compound body should fall faster than the heavy body, not slower.
Obviously, the long-held view that the heavier a body is, the faster it falls,
leads to an inescapable contradiction.
Galileo realized that the only logically acceptable view
was that all bodies, regardless of their weight
fall at exactly the same rate
once the effect of air resistance is removed.
Of course, if all bodies, in a vacuum, fall at the same rate,
the next question is:
exactly what is that rate?
From common experience,
we do know one thing about the rate of a falling body
the speed of a falling body increases as it falls,
which means that it accelerates,
dropping faster and faster as it falls.
Even before Galileo,
a number of scholars tried to formulate a description of this accelerated motion.
Some 100 years earlier,
Leonardo da Vinci made his own study of falling bodies,
driven perhaps by his dream of human flight.
Rather than ask how fast a body was falling,
da Vinci asked how far would it fall in successive intervals of time?
His theory of accelerated motion
was that a body would fall greater distances in later intervals.
He theorized that those distances would follow the integers,
that is, one unit of distance in the first time interval,
two units of distance in the second time interval, and so on.
Galileo himself adopted da Vinci's method of description
but he reached a different conclusion on how the distance increased.
Instead of increasing as the integers,
Galileo's theory was that in successive intervals of time,
the distances should follow the odd numbers.
Falling 1 unit of distance in the first time interval,
3 units of distance in the second interval,
5 units of distance in the third interval and so on.
In other words, according to Galileo,
the distance fallen is proportional to the odd numbers.
Galileo reached his conclusions after a brilliant series of experiments
in which he timed a ball as it rolled down steeper and steeper inclines,
moving closer and closer to the vertical path of a free-falling body.
Galileo's law of odd numbers
can be seen in action in a very unlikely place.
It's a place that would have amazed that great Renaissance thinker
even more than the surface of the Moon.
At Magic Mountain amusement park in Southern California,
customers gladly pay for the privilege of plummeting through space
under the influence of gravity.
Actually, that part of the ride is free.
What the customers have really paid for
is an arrangement that allows them to survive.
And anyway, what about Galileo?
If this is one unit of distance,
this should be three,
this should be five, and so on,
which is exactly what they are. Galileo was right.
In successive intervals of time,
the distances fallen do follow the odd numbers.
But there's something else going on here that Galileo understood perfectly.
Notice the total distance fallen at each point.
After the first time interval, 1 unit of distance.
After the second interval, 4 units of distance.
After the third interval, 9 units.
After the fourth, 16 units.
In other words, at the end of each interval,
the total distance fallen is
1, 4, 9, 16, 25 and so on.
And those numbers of course are the perfect squares,
so the distance fallen is proportional to the square of time.
And in that form, Galileo's law can be written as a simple equation.
Using s for distance, and t for time.
s(t) = ct²
This means we're talking about distance as a function of time.
the distance s increases as the square of time t²
This constant c is numerically equal to the distance a body falls in the first second.
That's 16ft, or just a little under 5m.
We know that at any point in the fall,
the distance fallen is equal to c times the square of time.
So after 2 seconds, the distance fallen equals c times 2² or 4c
If we use 16 for c, we know they've fallen 64ft.
Again, this symbol emphasizes that for any time t we can find the value of s.
At this point, even the most petrified freefall rider can depend on us
to tell her exactly how far she's fallen at each instant during the plunge.
But, the more discerning rider may also want to know how fast she's falling.
Her speed is the distance she falls divided by the time it takes.
For example, since she falls 64ft during the first 2 seconds,
her average speed must be 32 feet per second.
But that's only her average speed during the first 2 seconds.
At the beginning, she was standing still.
And at the end of 2 seconds, she was falling much faster than 32ft/s.
Obviously, what this woman really wants to know
is not her average speed
but her exact or instantaneous speed at any given time.
However,
if we try to use the same equation dividing the change in distance by the change in time,
we have a serious problem.
At any instant during the fall,
let's say at exactly 1.5 seconds,
the change in distance and time is zero.
So, a formula that determines speed
by dividing the change in distance between point A and point B by the change in time,
is of little use when we have a point A but no separate point B to work with.
To make matters worse,
both the top and the bottom of the fraction would be zero.
And of course dividing by zero is a mathematical disaster.
At first glance,
perhaps the expression "instantaneous speed" is a contradiction in terms?
And yet, common sense tells us that as long as an object is moving,
it must have a certain speed at every instant.
The problem is much more than a clever play on words.
It's a dilemma that plagued mathematicians for thousand of years.
But there is a way to solve it.
Instead of asking the instantaneous speed at an exact time t, we'll ask
What is the woman's average speed between time t
and a point h seconds later, at time t+h?
Now, the change in time is simply h seconds.
If the distance fallen at any time t equals c times t²
then the distance fallen at time t+h must equal c∙(t+h)²
The problem is solved.
We can calculate her average speed,
starting at any time t over any interval h.
h can be 1 second,
half a second, a tenth of a second,
or even zero.
Because now we're not dividing by zero.
And now we can let the h interval shrink smaller,
and smaller, and smaller,
even to the ultimate limit.
And at that instant, we've calculated a derivative
as the interval completely shrinks to zero.
If h is exactly zero,
we have found that at any time t,
her instantaneous speed, which we'll call v, is 2ct.
Using the value of 16 for c,
we can now tell her: "Madam, don't worry about a thing!"
The distance you've fallen is 16t² ft,
and your speed at each instant is simply 32t ft/s.
Obviously she's impressed.
"How did you figure all that out?", she might ask.
"It was nothing really. All we had to do was to invent the derivative."
In common usage, the word "derivative" means "arises from",
as in the phrase "fudge is a derivative of chocolate".
But in mathematics, the word has an exact technical meaning which amounts to this:
it's the rate at which something is changing.
The speed of the falling lady was the derivative of her distance from the top.
In other words, speed is the derivative of distance.
At first, when we discussed her average speed,
we were merely doing algebra,
simply plugging numbers into the speed-equals-distance-divided-by-time equation.
But when we began to work with an interval of duration h,
and at the right moment, let h shrink to zero,
we were calculating a derivative,
and we entered the world of differential calculus.
Differential calculus is the mathematics of using derivatives.
The process of calculating a derivative, is called "differentiation".
Of course, the concept of a derivative doesn't apply only to a body in motion.
Conceivably, a derivative could be calculated that represents
the rate of change in the population density of dolphins versus the temperature of the ocean.
Or the rate of change in the volume of a balloon versus its surface area,
or the rate of change in the cost of a pizza versus its diameter.
In other words,
a derivative can be calculated for almost any situation
in which one quantity changes as another quantity increases or decreases.
To get from distance to speed, we calculated a derivative.
But, what about the acceleration of a falling body?
To get from speed to acceleration, we do the same thing all over again.
If v, as a function of t, equals 2ct
then v of (t+h) equals 2c·(t+h)
a of t equals 2c, but look at what's happened.
First, we found that the distance s keeps increasing.
It depends on time: if t changes, s changes.
The speed v also keeps increasing with time.
But now we found that the acceleration a doesn't depend on time at all,
it's simply a constant, a equals 2c
Regardless of the value of t, a is always the same.
We've finally done it.
We've figured out that the result of gravity is constant acceleration.
We set out to answer 3 questions about a falling body.
How far? How fast? And how fast is it getting faster?
How far we found out pretty easily, just by watching our falling lady.
We even found her average speed, just by using algebra.
But to find out precisely how fast a body goes at each instant,
and to find out how fast it gets faster,
we needed our marvelous new mathematical tool,
the Derivative.
Using the derivative,
we have discovered the most elegant way to describe falling motion.
Bodies fall with constant acceleration.
Because that acceleration is so important, it has its own symbol:
a small g.
And g is equal to 2c.
Now we can put all 3 statements of the law of falling bodies in their final form
by replacing c with one half g
According to the law of falling bodies,
a body falls with constant acceleration,
with speed proportional to time,
and falls of distance proportional to the square of time.
That kind of motion is called "uniformly accelerated motion".
It is difficult,
but not quite impossible to discover all of these facts about uniformly accelerated motion
without using differential calculus.
And yet, Galileo understood all of these facts.
In fact, nearly 300 years before Galileo,
a French scholar named Nicole Oresme
had worked out the behavior of uniformly accelerated motion.
Oresme and Galileo used nearly identical mathematical methods to analyze the problem.
Their methods were based not on algebraic equations,
but on proportions between quantities, and on geometric figures.
The derivative was invented a generation after Galileo's death
by Sir Isaac Newton, and Gottfried Wilhelm von Leibniz.
With this powerful new method of analysis,
even more complicated kinds of motion could easily be analyzed.
Describing uniformly accelerated motion
became positively simple.
Without derivatives, it's difficult to understand what acceleration means,
much less describe uniformly accelerated motion
and work out all of its consequences.
And yet, that's exactly what Oresme and Galileo did.
They described uniformly accelerated motion
and worked out all of its consequences.
It was an act of sheer genius.
One of the jobs of physics is to find simple, economical underlying principles
to explain the complicated world that we live in.
We've done that today.
If I drop a body, it falls under the influence of the Earth's gravity.
As it falls, its motion is opposed with varying degrees of success
by the air through which it must fall.
If I can imagine disposing of the air,
and letting the body fall in vacuum,
then I discover a dramatic and surprising fact:
all bodies fall at the same rate.
I could be satisfied with that fact.
After all, discovering it was quite an impressive accomplishment.
But of course we're not satisfied,
we want to know "why is it true?"
What is the nature of gravity that leads to such strange behavior?
That question has turned out to be one of the deepest in all the history of physics.
It persisted even into our own century.
It was the starting point from which Albert Einstein built his general theory of relativity.
But we're getting ahead of our story.
Once we learned there was one law for all falling bodies,
the job was then to express that law with precision.
We've done that, too.
The law is:
All bodies fall with the same constant acceleration.
Acceleration is the rate of change of speed,
and speed is the rate of change of distance.
So we have in fact
3 precise mathematical statements of the law of falling bodies.
They're all true,
and they are related to each other by one of the great and crucial discoveries in the history of mathematics:
Differential Calculus.
The calculus was discovered by Isaac Newton and Gottfried von Leibniz.
It was a mighty triumph,
the most important event in mathematics in thousands of years.
Newton and von Leibniz
sacrificed the joy of their discovery
in a bitter dispute over who deserved credit for discovering it first.
All of these are threads in the story we're going to see unfold.
According to the law of falling bodies,
a body falls with constant acceleration,
at a speed proportional to time,
and falls a distance proportional to the square of time.
Subtitles by MonteCristo - 2012. Edited by Tran Nguyen Phuong Thanh - 2013.