[Music] annenburg [Music] [Music] media [Applause] [Music] [Applause] [Music] 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 it 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 Albert Einstein that's been one of the central mysteries in all of physics furthermore it says that bodies fall with constant acceleration it's almost impossible even to understand what that means without a marvelous mathematical device called a derivative we'll see today what that means and finally profound and important the all this is it violates our simplest intuition because it happens only in a vacuum not in the world we're familiar [Music] with for all of us the effect of the Earth's
gravity was probably our first encounter with the laws of nature [Music] 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 pool 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 exact L as we'd 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 remaining 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 [Music] rate [Music] be 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 well in my left hand I have a a feather in my right hand a hammer and I guess one of the reasons uh we got here today was
because of a gentleman named Nam Galileo a long time ago who made a rather significant discovery about falling objects in gravity fields and we thought that uh where would be a better place to confirm his uh findings and on the moon and I'll drop the two of them here and hopefully they'll hit the ground at the same time how about that proves that 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 [Music] [Music] 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 effective 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 [Music] flight rather than ask how fast a body was falling DaVinci 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 DaVinci'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 one unit of distance in the first time interval three units of distance in the second interval five 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 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 [Music] California customers gladly pay for the privilege of plummeting through space under the influence of gravity I'll buy you lots
of cotton candy we can feed the Ducks no I don't think I want to do this oh no actually that part of the ride is [Music] free we go what the customers have really paid for okay is an arrangement that allows them to survive at any rate what about [Music] gal 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 one unit of distance after the second interval four 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 of T = ct^ 22 this means we're talking about distance as a function of time the distance s increases as the square of time t^2 this constant C is numerically equal to the distance a body falls in the first second that's 16 ft or just a little under 5 m we know that at any point in the fall the distance Fallen is equal to C * the square of the time so after 2 seconds the distance Fallen equals c * 2^ 2 or 4 C if we use 16 for C we
know that they Fallen 64 ft again this symbol emphasizes that for any time T we can find the value of s at this point even the most petrified Free Fall Rider can depend on us to tell her exactly how far she has fallen at each instant during the plunge what if you 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 64 ft during the first 2 seconds her average speed must be 32 ft per
second but that's only her average speed during the first two seconds at the beginning she was standing still and at the end of two seconds she was falling much faster than 32 ft per second obviously what this woman really wants to know is not her average speed but her exact or instantaneous speed at any given time I'm fine 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 thousands 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 = C * T ^2 then the distance Fallen at time t +
H must equal C * t + h^ [Music] 2 [Music] [Applause] [Music] w 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 2 ct using the value of 16 te for SE we can now tell her Madam don't worry about a thing the distance you've Fallen is 16 * t^2 ft and your speed at each instant is simply 32 * T ft per second 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 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 ided 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 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 2 ct then V of t + h = 2 C * t + h [Music] a of t equal 2 C 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 2 C regardless of the value of T A is always the same we finally done it we figured out that the result of gravity is constant acceleration we set out to answer three 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 Al 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 2 C now we can put all three statements of the law of falling bodies in their final form by replacing C with 1/2 G [Music] according
to the law of falling bodies a body falls with constant acceleration with speed proportional to time and Falls a 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 [Music] facts in fact nearly 300 years before Galileo a French scholar named Nicole or had worked out the behavior of uniformly accelerated motion or 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 [Music] figures the derivative was invented a generation after Galileo's death by Sir Isaac Newton and gried Wilhelm van liit 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 or 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 theall 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 a starting point from which Albert Einstein built his general theory of relativity but we are 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 have 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 three precise mathematical statements of the law of falling bodies they are 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 gotfried V lietz it was a mighty
Triumph the most important event in mathematics in thousands of years Newton and Von liet sacrificed the joy of their Discovery in a bitter dispute over who deserve credit for discovering it first all of these are threads in the story we're going to see [Music] unfold according to the law of falling bodies a body falls with constant acceleration at a speed proportional to time and Falls the distance proportional to the square of time just an ordinary ordinary [Music] pump for sure [Music] anenberg media for information about this and other anenberg media programs call 1 1800 learner
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