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It was discovered by Galileo
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refined by Isaac Newton
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and in the hands of Albert Einstein,
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provided a theory of the mechanics of the cosmos.
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It was one of the deepest mysteries in all of physics.
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All bodies fall
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with the same constant acceleration.
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In a vacuum, all bodies fall
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with the same constant acceleration.
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That's it.
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That's the law of falling bodies.
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Doesn't seem like much to get excited about.
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And yet, just look at what it says.
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First of all, it says that the effect of gravity on all bodies
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is the same, regardless of their weight.
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From Galileo to Isaac Newton, right down to Einstein,
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that's been one of the central mysteries in all of physics.
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... says that bodies fall with constant acceleration.
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It's almost impossible, even to understand what that means
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without a marvelous mathematical device called the Derivative.
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We'll see today what that means.
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And finally, profound and important to all this is,
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it violates our simplest intuition
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Because it happens only in a vacuum
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not in the world we're familiar with.
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For all of us,
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the effect of the Earth's gravity was probably
our first encounter with the laws of nature.
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And whether or not we understand how gravity works
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we have an innate fear of what it does.
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But exactly what is the effect of gravity?
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Some bodies fall to the Earth quickly and directly
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while others behave quite differently.
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In some cases,
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the pull of gravity can be resisted almost indefinitely.
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To make any sense at all about how and why bodies fall
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we need to separate the effect of gravity on a falling body,
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from the opposing effect of the air through which the body is falling.
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In other words,
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we have to imagine a body falling not through the air
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but through a vacuum.
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For instance,
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what happens if a penny and feather
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drop simultaneously from the same height?
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They behave exactly as we would expect
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each falling at a very different rate than the other.
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But that's only because of the effect of air resistance on the two objects.
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In a vacuum, a penny, a feather, and any other object,
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will fall at exactly the same rate.
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With virtually no air remained inside the glass tube,
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the penny and feather are now in a vacuum.
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When the penny and feather are released,
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we'll witness the law of falling bodies in action.
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Without the effect of air resistance, in other words in a vacuum,
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all bodies, regardless of their weight,
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will fall at exactly the same rate.
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When Apollo 15 astronaut David Scott
explored the airless surface of the Moon,
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he couldn't resist repeating this classic
experiment for all the world to see.
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"Here in my left hand I have a feather,
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in my right hand a hammer,
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and I guess one of the reasons we got here today
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was because of a gentleman named Galileo a long time ago,
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who made a rather significant discovery
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about falling objects in gravity fields.
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And we thought that where
would be a better place to confirm his findings
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than on the Moon?
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And I'll drop the two objects and hopefully,
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they'll hit the ground at the same time.
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How about that?
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Looks like Mr. Galileo was correct in his findings."
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Mr. Galileo was correct.
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Nearly 400 years ago,
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at a time when all the world believed that
heavy bodies fall faster than lighter ones,
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Galileo realized that in a vacuum,
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all bodies should fall at the same rate.
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Galileo couldn't produce a vacuum,
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but he could imagine one.
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He pictured a heavy body attached to a lighter one.
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Would this compound body, he asked,
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fall faster or slower than the heavy body alone?
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If the lighter body did fall more slowly,
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it should slow down the heavy body.
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So the compound body should fall
more slowly than the heavy body alone
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But the compound body is actually heavier than the heavy body alone,
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therefore, the compound body should fall
faster than the heavy body, not slower.
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Obviously, the long-held view that the heavier a body is, the faster it falls,
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leads to an inescapable contradiction.
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Galileo realized that the only logically acceptable view
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was that all bodies, regardless of their weight
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fall at exactly the same rate
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once the effect of air resistance is removed.
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Of course, if all bodies, in a vacuum, fall at the same rate,
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the next question is:
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exactly what is that rate?
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From common experience,
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we do know one thing about the rate of a falling body
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the speed of a falling body increases as it falls,
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which means that it accelerates,
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dropping faster and faster as it falls.
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Even before Galileo,
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a number of scholars tried to formulate
a description of this accelerated motion.
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Some 100 years earlier,
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Leonardo da Vinci made his own study of falling bodies,
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driven perhaps by his dream of human flight.
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Rather than ask how fast a body was falling,
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da Vinci asked how far would it fall
in successive intervals of time?
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His theory of accelerated motion
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was that a body would fall greater distances in later intervals.
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He theorized that those distances would follow the integers,
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that is, one unit of distance in the first time interval,
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two units of distance in the second time interval, and so on.
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Galileo himself adopted da Vinci's method of description
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but he reached a different conclusion on how the distance increased.
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Instead of increasing as the integers,
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Galileo's theory was that in successive intervals of time,
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the distances should follow the odd numbers.
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Falling 1 unit of distance in the first time interval,
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3 units of distance in the second interval,
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5 units of distance in the third interval and so on.
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In other words, according to Galileo,
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the distance fallen is proportional to the odd numbers.
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Galileo reached his conclusions
after a brilliant series of experiments
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in which he timed a ball as it
rolled down steeper and steeper inclines,
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moving closer and closer to the vertical path of a free-falling body.
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Galileo's law of odd numbers
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can be seen in action in a very unlikely place.
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It's a place that would have amazed
that great Renaissance thinker
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even more than the surface of the Moon.
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At Magic Mountain amusement park in Southern California,
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customers gladly pay for the
privilege of plummeting through space
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under the influence of gravity.
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Actually, that part of the ride is free.
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What the customers have really paid for
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is an arrangement that allows them to survive.
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And anyway, what about Galileo?
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If this is one unit of distance,
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this should be three,
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this should be five, and so on,
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which is exactly what they are. Galileo was right.
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In successive intervals of time,
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the distances fallen do follow the odd numbers.
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But there's something else going on here
that Galileo understood perfectly.
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Notice the total distance fallen at each point.
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After the first time interval, 1 unit of distance.
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After the second interval, 4 units of distance.
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After the third interval, 9 units.
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After the fourth, 16 units.
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In other words, at the end of each interval,
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the total distance fallen is
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1, 4, 9, 16, 25 and so on.
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And those numbers of course are the perfect squares,
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so the distance fallen is proportional to the square of time.
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And in that form, Galileo's law
can be written as a simple equation.
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Using s for distance, and t for time.
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s(t) = ct²
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This means we're talking about distance as a function of time.
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the distance s increases as the square of time t²
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This constant c is numerically equal
to the distance a body falls in the first second.
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That's 16ft, or just a little under 5m.
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We know that at any point in the fall,
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the distance fallen is equal to c times the square of time.
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So after 2 seconds, the distance fallen equals c times 2² or 4c
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If we use 16 for c, we know they've fallen 64ft.
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Again, this symbol emphasizes that
for any time t we can find the value of s.
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At this point, even the most petrified freefall rider can depend on us
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to tell her exactly how far she's fallen at each instant during the plunge.
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But, the more discerning rider may also
want to know how fast she's falling.
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Her speed is the distance she falls divided by the time it takes.
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For example, since she falls 64ft during the first 2 seconds,
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her average speed must be 32 feet per second.
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But that's only her average speed during the first 2 seconds.
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At the beginning, she was standing still.
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And at the end of 2 seconds,
she was falling much faster than 32ft/s.
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Obviously, what this woman really wants to know
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is not her average speed
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but her exact or instantaneous speed at any given time.
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However,
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if we try to use the same equation
dividing the change in distance by the change in time,
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we have a serious problem.
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At any instant during the fall,
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let's say at exactly 1.5 seconds,
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the change in distance and time is zero.
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So, a formula that determines speed
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by dividing the change in distance between
point A and point B by the change in time,
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is of little use when we have a point A
but no separate point B to work with.
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To make matters worse,
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both the top and the bottom of the fraction would be zero.
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And of course dividing by zero is a mathematical disaster.
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At first glance,
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perhaps the expression "instantaneous speed"
is a contradiction in terms?
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And yet, common sense tells us that as long as an object is moving,
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it must have a certain speed at every instant.
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The problem is much more than a clever play on words.
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It's a dilemma that plagued mathematicians
for thousand of years.
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But there is a way to solve it.
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Instead of asking the instantaneous
speed at an exact time t, we'll ask
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What is the woman's average speed between time t
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and a point h seconds later, at time t+h?
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Now, the change in time is simply h seconds.
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If the distance fallen at any time t equals c times t²
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then the distance fallen at time t+h must equal c∙(t+h)²
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The problem is solved.
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We can calculate her average speed,
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starting at any time t over any interval h.
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h can be 1 second,
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half a second, a tenth of a second,
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or even zero.
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Because now we're not dividing by zero.
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And now we can let the h interval shrink smaller,
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and smaller, and smaller,
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even to the ultimate limit.
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And at that instant, we've calculated a derivative
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as the interval completely shrinks to zero.
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If h is exactly zero,
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we have found that at any time t,
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her instantaneous speed, which we'll call v, is 2ct.
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Using the value of 16 for c,
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we can now tell her: "Madam, don't worry about a thing!"
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The distance you've fallen is 16t² ft,
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and your speed at each instant is simply 32t ft/s.
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Obviously she's impressed.
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"How did you figure all that out?", she might ask.
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"It was nothing really.
All we had to do was to invent the derivative."
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In common usage, the word "derivative" means "arises from",
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as in the phrase "fudge is a derivative of chocolate".
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But in mathematics, the word has an exact
technical meaning which amounts to this:
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it's the rate at which something is changing.
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The speed of the falling lady was
the derivative of her distance from the top.
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In other words, speed is the derivative of distance.
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At first, when we discussed her average speed,
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we were merely doing algebra,
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simply plugging numbers into the
speed-equals-distance-divided-by-time equation.
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But when we began to work with an interval of duration h,
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and at the right moment, let h shrink to zero,
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we were calculating a derivative,
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and we entered the world of differential calculus.
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Differential calculus is the mathematics of using derivatives.
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The process of calculating a derivative, is called "differentiation".
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Of course, the concept of a derivative
doesn't apply only to a body in motion.
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Conceivably, a derivative could be calculated that represents
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the rate of change in the population density
of dolphins versus the temperature of the ocean.
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Or the rate of change in the volume of a balloon versus its surface area,
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or the rate of change in the cost of a pizza versus its diameter.
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In other words,
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a derivative can be calculated for almost any situation
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in which one quantity changes as
another quantity increases or decreases.
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To get from distance to speed, we calculated a derivative.
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But, what about the acceleration of a falling body?
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To get from speed to acceleration,
we do the same thing all over again.
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If v, as a function of t, equals 2ct
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then v of (t+h) equals 2c·(t+h)
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a of t equals 2c, but look at what's happened.
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First, we found that the distance s keeps increasing.
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It depends on time: if t changes, s changes.
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The speed v also keeps increasing with time.
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But now we found that the acceleration a doesn't depend on time at all,
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it's simply a constant, a equals 2c
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Regardless of the value of t, a is always the same.
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We've finally done it.
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We've figured out that the result of gravity is constant acceleration.
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We set out to answer 3 questions about a falling body.
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How far? How fast? And how fast is it getting faster?
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How far we found out pretty easily, just by watching our falling lady.
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We even found her average speed, just by using algebra.
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But to find out precisely how fast a body goes at each instant,
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and to find out how fast it gets faster,
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we needed our marvelous new mathematical tool,
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the Derivative.
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Using the derivative,
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we have discovered the most elegant way to describe falling motion.
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Bodies fall with constant acceleration.
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Because that acceleration is so important, it has its own symbol:
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a small g.
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And g is equal to 2c.
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Now we can put all 3 statements of
the law of falling bodies in their final form
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by replacing c with one half g
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According to the law of falling bodies,
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a body falls with constant acceleration,
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with speed proportional to time,
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and falls of distance proportional to the square of time.
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That kind of motion is called "uniformly accelerated motion".
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It is difficult,
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but not quite impossible to discover all of
these facts about uniformly accelerated motion
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without using differential calculus.
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And yet, Galileo understood all of these facts.
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In fact, nearly 300 years before Galileo,
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a French scholar named Nicole Oresme
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had worked out the behavior of uniformly accelerated motion.
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Oresme and Galileo used nearly identical
mathematical methods to analyze the problem.
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Their methods were based not on algebraic equations,
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but on proportions between quantities, and on geometric figures.
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The derivative was invented a generation after Galileo's death
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by Sir Isaac Newton, and Gottfried Wilhelm von Leibniz.
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With this powerful new method of analysis,
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even more complicated kinds of motion could easily be analyzed.
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Describing uniformly accelerated motion
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became positively simple.
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Without derivatives, it's difficult to
understand what acceleration means,
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much less describe uniformly accelerated motion
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and work out all of its consequences.
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And yet, that's exactly what Oresme and Galileo did.
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They described uniformly accelerated motion
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and worked out all of its consequences.
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It was an act of sheer genius.
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One of the jobs of physics is to find
simple, economical underlying principles
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to explain the complicated world that we live in.
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We've done that today.
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If I drop a body, it falls under the influence of the Earth's gravity.
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As it falls, its motion is opposed with varying degrees of success
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by the air through which it must fall.
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If I can imagine disposing of the air,
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and letting the body fall in vacuum,
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then I discover a dramatic and surprising fact:
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all bodies fall at the same rate.
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I could be satisfied with that fact.
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After all, discovering it was quite an impressive accomplishment.
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But of course we're not satisfied,
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we want to know "why is it true?"
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What is the nature of gravity that leads to such strange behavior?
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That question has turned out to be one
of the deepest in all the history of physics.
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It persisted even into our own century.
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It was the starting point from which
Albert Einstein built his general theory of relativity.
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But we're getting ahead of our story.
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Once we learned there was one law for all falling bodies,
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the job was then to express that law with precision.
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We've done that, too.
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The law is:
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All bodies fall with the same constant acceleration.
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Acceleration is the rate of change of speed,
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and speed is the rate of change of distance.
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So we have in fact
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3 precise mathematical statements of the law of falling bodies.
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They're all true,
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and they are related to each other by one of the great
and crucial discoveries in the history of mathematics:
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Differential Calculus.
335
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The calculus was discovered by
Isaac Newton and Gottfried von Leibniz.
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It was a mighty triumph,
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00:26:50,300 --> 00:26:54,580
the most important event in mathematics in thousands of years.
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Newton and von Leibniz
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sacrificed the joy of their discovery
340
00:26:59,040 --> 00:27:03,420
in a bitter dispute over who deserved credit for discovering it first.
341
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All of these are threads in the story we're going to see unfold.
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According to the law of falling bodies,
343
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a body falls with constant acceleration,
344
00:27:18,020 --> 00:27:21,840
at a speed proportional to time,
345
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and falls a distance proportional to the square of time.
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Subtitles by MonteCristo - 2012.
Edited by Tran Nguyen Phuong Thanh - 2013.