Program 14 - "The New Physics: Freefall and Inertia"

Music Hey, come on guys, I really can do this you know.I'll get it.I'll get it right.I really will.It hasn't been that many tries.Come on, Russell, are you there?James, Allen, Nick, is anybody there?Come on guys, we can do this, I know we can.It won't take that long.I've done it before.It's only been 500 times.We can do this, I know we...MusicSilico: "We are back with Science 122, the Nature of Physical Science.This is the telecourse that adds time to the universe.This is program four-teen, lesson two point six.We call it, "The NewFizz-ickxz."Before we're done with this lesson we will have seen howGalileo tested his theories of freefall and other motion usinga combination of inductive and deductive logic togetherwith observations and precisely designed experiments.Along the way we will see how Galileo made assumptions,tested suppositions, made inferences and drew conclusionsas he rewrote the book of motion, proving Aristotle's world to benonsense, and forging a new paradigm in the process.We will look at a creative mind making an intuitive leap as wesee how Galileo used the results of his experiments and logic tosolve the age-old problem of projectile motion.

Here are the objectives for today's lesson.These objectives are also in the Study Guideat the beginning of the lesson.Before you begin to study the lesson, take a few minutesto read the Objectives and the Study Questions for this lesson.Look for key words and ideas as you read.Use the Study Guide and follow it as you watch the program.Be sure to read these Objectivesin the Study Guide and refer to them as you watch the lesson.Focussing on the learning objectives will help youto study and understand the important concepts.Compare the Objectives with the Study Questions for the lessonto be sure that you have the concepts under control.You know, Newton's gravitational act is a hard one to follow,so I thought I'd better get my own gravitational act working first.Silico: "Do not quit the day job."Speaking of day jobs, do you like yours?Huh, boy that new system upgrade!You don't know what's in those things.Well, I suppose we have to get to the program eventually.In the previous program we tried to establish someinsight into Galileo and his times.We also learned about the different tools that Galileo used in his work.

Now it's time to see how he used those toolsto analyze motion and to ascertain its true nature.The goal of disproving Aristotle's viewson motion, Galileo met quite successfully.Through his use of logic, mathematics, experimentsand observations he not only refuted Aristotle, but alongthe way he made some other surprising discoveriesof the properties of matter and motion and in the processset the stage for Newton's work on gravity and the subsequentdevelopment of the conservation laws.Even today Galileo's discoveries define ourunderstanding of the nature of motion.In addition to his motion studies, and his telescopic observations,Galileo used logic and mathematics in ways that hadnot been tried before to discover relationships which werequantitatively testable in a laboratory.He defined motion in an unambiguous way by introducingthe concept of time and its relationship with space or location.From the data he collected in very cleverly designed and carefullycontrolled and repeated experiments, he was ableto show that pure freefall acceleration is uniformand constant for all objects regardless of their own size and weight.

The air, regarded by Aristotle as the controlling factor in naturalmotion, was shown by Galileo to bean interference to motion rather than the cause of it.After observing the motion of balls of various textures asthey rolled on inclined planes of various angles, he concludedthat without gravity to speed the descent and slow the ascent,and without friction, objects would not start or stop moving at all.Recognizing that falling objects accelerate downwardat the same uniform rate even if they are moving horizontallyallowed him to describe the motions of projectiles.Galileo's description of projectile motion as a combinationof horizontal and vertical motion, in the same terms and with thesame kinds of relationships destroyed Aristotle'sconcept of different kinds of motion.Through his efforts, and despite the political suppressionof his ideas and writings, Galileo established a new paradigm of motion.In this new view, the idea that motion ceased unless activelymaintained was replaced by the notion that motioncontinued unless it was interfered with.

Galileo used a number of methods to attack Aristotle's conjecturethat objects freefall at a rate proportional to their gravity,or heaviness, and that their speed is controlledby the medium through which they're falling.Recall that Aristotle thought thatfreefall motion, which he called natural motion,was caused by the desire for a substance to attainits rightful place in this concentric hierarchy of matter.So, according to Aristotle, the heavier an object was, the moredesire it had, and, therefore, the greater necessity to find thatplace, and, therefore, the faster it would fall, or rise in the caseof lightness, like air bubbles in water or fire in air.To counter Aristotle, while at the same time appealing to the oldparadigm, Galileo applied the logical dialogues of Plato, usingboth inductive and deductive logic to theorize aboutand observe the behavior of objects in freefall.He applied his own methods of mathematical generalizationto derive new relationships from precise definitions, and we sawthose definitions in the previous lesson,and, therefore, described motion as a rate of change.And he also performed very controlled experimentsrepetitively and analyzed the data in order to compare it to his predictions.But, of all the methods used by Galileo, it was his abilityto creatively generalize and extrapolate which led to the real discoveries.

Imagine the feeling that he must have had to not only disprove2000 years of wrong thinking, but also to see how simple itbecomes when certain verifiable assumptions are made.The key word here is "verifiable."Why is that a significant word?What is it about verifiable that makes the difference?OK, so Galileo's logic did not prove conclusively thatAristotle's freefall was incorrect.It only showed it to be logically inconsistent.That's all it takes logically, but as Aristotle unknowinglydemonstrated to us, logic alone is not enough.

Logic is enough to cast serious doubt, but in his attemptto maintain logical consistency Aristotlehad painted himself into an intellectual corner.Using the same methods of discourse, but employing bothinductive and deductive logic in a complimentary way, Galileotook a very different approach to reality.He attacked the logic of Aristotle's conjectureabout natural motion being dependent upon weightand composition in a very, very clever and very insightful way."Suppose," Galileo asked "that you tie two weights of differentgravities or different heaviness together and then you drop them.And assuming that the rope holding them together does notstretch, then one of the following might occur."But the question is, which one?On one hand the heaviness of one weight interactswith the lightness of the other so they fall at a reduced ratewhich is somewhere in between the rate that either would fallalone; maybe one and a half times.OK?Or, the two objects as a unit have more gravity than eitherof them alone, so they fall at a faster rate than either would fall alone.Or, the lighter weight dilutes the whole thing and so they fallat the slower rate of the lighter weight.Or, they fall at the faster rate of the heavier weight.And finally Galileo's choice was, they fall at the samerate whether or not they're tied together.You see, if objects fall at a different rate according to theirgravity, then there's no clear way to predictwhich one of these might actually occur.

Now you'd think it would be easy to go outside and try it.So why do you think more people didn't?The only explanation which does not produce a seriesof paradoxical possibilities, is that all of the objects fallat the same rate regardless of their weight, and regardlessof whether they're tied together.Now consider this for a minute.If the two objects fall at the same rate, then they will do sowhether they're tied together or not.There is no ambiguity, no confusion.So, only this choice, that they all fall at the samerate, provides an answer which is not ambiguous.Now this is a brilliant and simple argument, which proves nothing.What it does is to suggest that it makes more sense if everythingfalls at the same rate, regardless of why it might be so.It also sets up the desire to actually watch things drop,which Galileo did, and he demonstrated this to others.Most of them thought his demonstrations to be nothingmore than parlor tricks, or thought that he was somehowbewitching them and making them see things that didn't happen.After all, that's what magic is, right?OK, now Galileo used other logical arguments as well.Suppose, for example, that you have a very large weight and a very small weight.

Now here we have a weight that's 1000 poundsand a weight that's only 1/1000 of a pound.So, in this particular scheme, the larger weightweighs a million times more than the other.Galileo asserted that it was preposterous to assume thatif one object weighed a million times more than the other,then it might reach the ground a million times faster.So, if you were dropping something like this onlyfrom the height of the top of your head, for example,the thousand pound object would reach to the groundbefore the lighter weight had even moved at all.Now, this certainly might be true for a marble and a grain of dust,but it's certainly not true for a marble and a bowling ball.So try it yourself if you have one.We've already noted in Program 13 how Galileoused both inductive and deductive logic to deriveand test specific relationships.We'll see more clearly in the next section how he did this when weactually study Galileo's experiments.For instance, you use deduction in the form of algebra.You start with definitions of velocity and uniformacceleration, then you derive a relationship between distanceand time when an object is uniformly accelerated.This is using principle to predict outcome.This is deductive logic.

Next, you design an experiment which will allow you to observethe behavior that you wish to observe; in this case,distance and time and their relationship.You observe and record the behavior and you have a bunch of numbers.Then what?Well, then you compare the numbers to see if they matchwhat your deduction predicted they should be.That's what we mean by deduction.So, what about induction?Well, easy.For induction, you observe that the ball rolls uphill after it rolls downhill.Then you use induction to discover a general rule thatapplies in both cases; that is uphill and downhill.In this case you decide that the principle is that the same causeapplies in both the uphill and the downhill case,so the amount of influence is same in both cases.In other words, the amount that the ball's acceleratedon the downhill part is the same amount that it'sdecelerated on the uphill part.

Now, you see how this is working together.You take this conclusion as a general rule and go deductivewith it again and then you predict that if your induction iscorrect, then the ball should roll uphill to the sameheight from which it was rolled down hill.Now you've deduced another testable behavior whichis true if your induced rule is true.So you see what you've done.You take the mathematics, you deduced from that a behavior.You see if that behavior is correct.If it is, then you use that to induce a general principle,and then you come up with that principle and use that principleto deduce yet another set of behavior which then you can go back and test.You see how it works?Now, you might need to read this preceding paragraph over acouple of times in the Study Guide to make sure that you've got it.If you don't see it, try drawing a diagram likethe one at the end of Lesson 13.You know that triangle with the experiment and induction and deduction?What about objects which roll downhill?Is that like or unlike freefall?In what ways is it different?For one thing, Galileo noted downhill motion is horizontaland vertical at the same time, kind of like a bugcrawling sideways on a moving board.This seemed contrary to Aristotle's notion that violentand natural motion were different in nature.

For another thing, an object rolling downhill gainsspeed at a slower rate than one in freefall.We've already seen in Lessons 12 and 13 the way in which Galileoused mathematics to define velocity, average velocity, acceleration.And then you may remember that I showed you on the ELMO how heused those definitions to derive a relationshipbetween distance and the second power of time.You remember that don't you?The equation of the formula "D" equals one half "A,""T" squared, and how we combined everything together that way?From those definitions Galileo derived the testable descriptionof uniformly accelerated motion in terms of the mathematicalrelationship between distance and time.The mathematical description as a rate of change, of motionI should say, as a rate of change was unheard in Aristotle's scheme.

No one ever thought about this before.This analysis led to the conclusion that in uniformly acceleratedmotion a linear relationship exists between distanceand the square of time, with the second power of time.If you're so inclined you may want to go back and look at thisanalysis and note that acceleration is the rate of a rate.Thus the appearance of time twice, as in time squared.The concept of a rate of change is abstract enough to have escapedthousands of years of great minds who thought about change.The concept of a rate of change of a rate of changeis one step further removed even than that.It's like moving from flatland to three dimensions.Like people who can't see three dimensions of flat drawings.Most of us have difficulty comprehending such things as a rate of a rate.

OK, so, although a rate of change of a rate of change is anabstract concept, there are other similar examples which are easier to grasp.Interest rates, for example, at the bank.Suppose you invest money in the bank at a time whenbanks are paying 3% interest.This is a rate of growth which will allow you to know howmuch your investment will be worth after a certain amountof years has passed, or a certain amount of time has passed.Suppose that you know that the rate of interest increases onehalf percent per year, so that the first year it pays 3% and thenext year it pays 3 and a half percent, and the following year 4%.This is a rate of change of a rate of change.Right?The interest rate is the rate at which yourmoney grows, and that interest rate, itself, changes.From this you could know two things.

First of all, you could know the interest rate in any yearfrom a simple addition of one half percent per year.Second, you could know how much the investment is worth at anytime because you know the interest rate at any time.So, even here, time is involved twice.One to calculate the interest rate and once again to calculate thevalue once the interest rate is known.So, if you feel mathematically challenged and have difficultywith the concept of a rate of a rate, note that the conceptof rate and changing rate are not just useful for an obscurecalculations of distance and time.And also feel a little bit better that it took peopleup to Galileo's time 2500 years of thinking about changebefore anybody came up with the idea.Equally important in this whole concept wasGalileo's use of time in the description of change.

Well, strangely enough time had not entered into previousnotions of motion, except in the sense of at what timea particular event would occur like the passage of a planetover the Zenith or something like that, clock time, that is.The idea of using time as a variable in a setof relationships was Galileo's alone, and we'll see in the nextprogram how Newton took this and defined, actually, the differencebetween absolute and relative and clock time and all that.We use time all the time and rely on it without anythought at all of its properties.Maybe that's why no one thought to use it in this way before Galileo.Time is an abstract concept.It's the concept that events happen not only in space, but also in time.It adds another dimension to our perspective when we alreadyhave enough trouble comprehending only three dimensions.So you might say from this that Galileo discovered timein the same way that the Greeks discovered the mind.

We'll explore these aspects of time bit by bit in upcominglessons, just because we're curious and also to remind usthat there may be other things like time that we tend to takefor granted without really understanding it.(Crunch.)It's time for another Food for Thought.(Clock ticking.)As a way to prepare our minds for the future, think and write about this one.What is time?Not, what time is it, but what is time?Try to describe its properties.Is it matter, or energy, or something else?Does it really exist, or is it just a convenience used to describechange the way Ptolemy's devices described the motions of the planets?Silico: "A voice in my microprocessor just saidtime is a nonspatial continuum in which eventsoccur in apparentlyirreversible successionfrom the past through the present to the future."Oh, that sounds good, but what does it mean?Silico: "I have no idea."As for the experiments, they were not just welldesigned in terms of the defined variables.They were specifically designed to test the theoreticalrelationships Galileo had so laboriously defined and derived.It works like this.

If a numerical relationship of the measured distances and timeswas of the same type; that is, linear relationshipbetween distance and the square of time, the same typeas the theoretically derived relationship, then it madea strong case that reality was actually definedin some general way by this relationship.Specifically the assumption is that if the relationshipbetween distance and time turns out to be as the mathematicspredicted, then it's safe to conclude that this relationshipdoes, indeed, apply and that the falling object is uniformly accelerated.So, just in case at this point you might be getting a senseof "So . . .," remember that this is not reallyabout balls rolling down hills, it's about truth and authority.Galileo's purpose in doing all this was to prove that Aristotle waswrong about motion because if Aristotle was wrongabout motion, then he could also be wrong about geocentrismand spherical perfection and all that sort of thing.Is the fact that it is the type of relationshipbetween the numbers and not the numbers themselves consistentwith Pythagorean beliefs or against it?Can you write a short essay that argues that it isPythagorean, but alsonon-Pythagorean?Can you argue both side convincingly?OK.

Another important aspect of these experiments wasthe number of repetitions that Galileo did.He recognized that the more measurements you takethe closer the results approximated reality.After hundreds of trials over several years, any irregularitiestend to be smoothed out and overwhelmed by the averages.If you only do one, you might make a mistake.If you only do two, you don't know which one is right.So the more you do the closer you come to actually getting the average value.But, the formalization of this concept of averagesand statistics was still 200 years away at this point, but we'll seeit again elsewhere in our travels downstreamin this river of scientific heritage.Aristotle had done experiments where he correctly induced thatstones and other smooth objects falling through water do indeedreach a terminal velocity that is proportional to their weight,or more appropriately, to their density.What Aristotle had failed to understand was that theextrapolation from water to air does apply, but that water ismillions, millions of times more thicker, that is, more viscous than air.So when objects fall through a medium such as air or waterthey accelerate until the effort required to move through themedium at a certain speed is great enough to matchthe weight of the object in the medium.In other words, the downward force of gravity is balancedby the upward force of the medium on the object.When that happens the object no longer accelerates, but theymove at a constant speed, which we call the terminal velocity.The time it takes for an object falling through water to reachterminal velocity is very small because waterretards the motion, very much, in fact.

In air, a very dense object such as lead weights are not affectedvery much by air resistance, and, in fact, will fall a greatdistance before reaching terminal velocity.Whereas in water a lead weight this big around might reach itsterminal velocity in a matter of a few seconds.That same lead weight this size falling through air might fallfor five minutes before it reaches its terminal velocity.So, the problem was then that over the short distancesof the laboratory the motion of very dense objectsmight not be affected much at all.If we can measure the distance that they fell in various timeintervals, we could analyze their motion to see if it is uniformly accelerated.The problem is that if we have a ball and we drop it.Here we go, let me drop it.Let's time how long it takes it to hit the ground or hit the table.Oh, look, it goes really fast.Let's try it again.The ball goes so fast, in fact, that when you drop things in air,everything happens so quickly and even if you had a stopwatch,which you can start and stop, you'd have a hard time timing itbecause your reflex time is generally shorter thanthe amount of time it takes the ball to fall.And even considering that, Galileo didn't have a stopwatch.

On the other hand, we have the opposite problem.If you drop the ball from heights great enough to time their fallwith any precision, it requires that they be in the air longenough for the air to significantly affecttheir velocity as their speed increases.

Do you see the problem?If you drop something a short distance, it falls too quickly to measure.If you drop it a long distance, the air interferes with itsmotion so you don't get a true picture of the motion.Aristotle came to the same conclusion but he solved theproblem by dropping things through water because hethought that the medium, that is, the wateror the air, was the prime factor in motion.His paradigm drove the way in which he tested the hypothesis,which was that things fall at constant speeds.Galileo chose a different approach.He decided to slow down the motion through the air in someway which did not affect the nature of the relationshipbetween distance and time.He correctly reasoned that a ball rolling downhill on a flatsurface moved downhill according to the same cause as a ballin freefall, regardless of what that cause was.

Galileo adopted Aristotle's terminology and simply saidthat the cause of this motion was the gravity of the ball.By this he meant what we would mean today, the weight of the ball.So, here we use the word gravity in the context of a qualityof the ball, as in "the ball falls because of its heaviness."Note that the cause of the motion here is independentof the quality of heaviness, which exists as a drivingforce for motion regardless of the cause.Here's one of those "wrong" questions that Galileo asked.Now Aristotle taught that the ball fell to attain its appropriateplace and that in the absence of a mediumsuch as air or water, it would fall at infinite speed.Now this is an interesting assumption.But is it a warranted assumption?What if air is not the controlling factor of motion at all,as Aristotle taught, but rather, suppose air is a minorhinderance to the motion which masks the truemotion rather than controlling it?Then what?Well, if we could get rid of the air, and just pump it awayso that we could see the pure uninterrupted motion before airhad a chance to alter it, then we know for sure.

The problem was that the vacuum pump would not be inventedfor another 150 years so they could not do the thingwith the tube that we will do here on the tube with the tube.This video was taken on the surface of the moon.One of the early manned missions to the moon included as partof the design experiments a hammer and a feather.The airless surface of the moon truly allowed this experimentto be tested for the first time in a nearly perfect vacuum.This experiment also demonstrated that the effectsof gravity on the moon were the same as those here on earth.Only just a little bit slower.You've got to figure that they spent all this moneyto send that hammer up there to the moon.It must have been a fairly important experiment.In fact, this was the first physics experiment ever done on a different planet.Well, how insightful of Galileo to be able to conclude that whichcould only be tested on arrival at the surface of another planet,earthlike in its gravity as well as in its mountains and craters.

So let's watch the video again and this time pay close attentionto the feather and the hammer falling together.Astronaut: "And so we thought we'd try it here for yah,and the feather happens to be appropriately a falcon featheror a falcon, and I'll drop the two of them hereand hopefully they'll hit the ground at the same time.How about that?"So even though Galileo couldn't suck the air out of his roomor out of a tube because he didn't have a vacuum pump, we do.What I have here is a tube from which I've sucked all the air out.Or actually I've attached the vacuum pump to it and drawn the air out.The little valve up here, the little thing that I can turn to keep the vacuum sealed.Inside the tube is a feather and a coin.I don't know if you can see those or not.There's the feather and there's the coin.You can probably even hear the coin rattling.So what I want to do is to turn the two upside down and see if thecoin and the feather really fall together.

OK, you ready?Here we go.This is going to happen fast so you gotta watch close.Ready?One, two three.(Clink.)You see that?It was kind of fast, let's do it again.OK, one, two, three.(Tick, clink)Hear the coin and the feather falling together?Even if you can't see the coin, you can hear it hit,at the same time that the feather does.One more time, one, two, three.(Tick, clink.)So now, I wonder if we should let the air outof this and see if it really works that way.So when I turn the valve you should be able to hear the air rushing back in.(Shhhh.)OK, so now we got the two filled up with air again.Let's try the same thing again.Make sure the feather isn't stuck someplace.It does that sometimes, so, I've lost my feather.(Clink, clink.)OK, well let's just turn it upside down a few times and see what happens.Ooops, there's a piece of the feather.Another piece of the feather.You know, I always wondered how you canlose a feather inside the closed tube.Excuse me for a minute.There it is.

OK, so you notice the feather fluttering down.Let me try turning it upside down now and watch what happens.You should notice this time that the coin reachesthe bottom much before the feather does.(Clink.)See how the feather sort of flutters down there.So what we're seeing here is that in the absence of air resistance,the objects, the coin and the feather, fall at the same rate.They accelerate at the same rate.Regardless of the fact that the coin weighs probably a hundredtimes more than the feather does.So in Galileo's time, he could not remove the air physically,but he could do it mentally through the use of a logical inference.We'll get into this in more detail as we study the experiments.With this idea, Galileo also, unknowingly, introduced thenotion of limits which Newton would use to great advantagein analyzing planetary motions 60 years later.The concept of a limit is easy in principle although difficult in practice.Suppose you start with a lot of interference like water and thenyou gradually reduce it noting the effect on motion.And then you extrapolate the results to the case of no resistance.Simple, no?Not really.It's really a difficult concept at first.Certainly it's a difficult one to have thought of in the first place.What a concept.No wonder it took people so long to figure this stuff out.So, the idea is you minimalize and generalizeto the ideal case of no air at all.Only then can the true nature of motion be understood.Unless, of course, you can go to the moon.Later the effects of the air and other resistance to motion can be studied.

Now, do you see how this is an exampleof model-building.It's like wiping the dust off an old photographor cleaning the patina off the silverware.The model-building process tries to find the essenceof the problem by removing those things which are not essential to its understanding.Galileo's insight and his intuition into the nature of motionallowed him to recognize the true role of air resistanceand its similarity to rolling resistance.This process, by the way, of breaking the motion into partsor smaller problems and then solving them separatelyis a form of reductionism which Galileo originatedand it ultimately led to the writing of algorithmsin construction of computer codes which drive the program whichwe are using to produce this television program.Galileo sought a practical method to minimize the air resistancewhile maximizing the time over which the measurement couldbe made and at the same time controlling the amountof interference encountered by the moving objects.The problem of time was a serious one.Galileo actually discovered in church one Sunday morning,I suppose he was bored, had nothing else to do, that theswinging chandelier in the church took the same amount of timefor one swing regardless of how widely it was swinging.

Now this is an unexpected result and it's one that you might not expect.So if a pendulum is swinging way back and forth like thisor making a little swing like this, it still takes the same amount of time.He used his heartbeat to verify this.Now this is an assumption that your heart beats at the samerate, and I'm sure that when he discovered this his heartmight have speeded up a little bit.Some scholars have claimed that this is what gave him the ideato use time as a variable in the first place.We'll probably never know for sure, but we do know thathe used the pendulum as a steady timing devise insteadof a heartbeat, which he tried first and it didn't work very well.Because the heartbeat is variable dependingon the mood and the state of exertion that you're in.Although Galileo later used the more precise clock madeof water, his recognition and use of the pendulum as atimekeeping device stimulated the development of mechanicalclocks and initiated the practice of accurate timekeeping.This alone, in the absence of Galileo's other contributionswould have had a profound impact on history.Later on, by the way, in Holland, Christopher Huygens woulddesign an accurate mechanical clock, the pendulum clock,based on Galileo's observation which would revolutionizethe science of physics, as well as the sciences of navigation and astronomy.

A century and a half later, the accuracy of the clockand Newton's law of gravity would motivate the voyage of Captain James Cook.This voyage, of course, which brought European cultureto the Hawaiian islands for the first time, was promptedby the desire to verify Newton's laws with the help of anextremely accurate pendulum clock.With the idea that motion down an inclined plane is both horizontaland vertical, Galileo saw that the flat, sloping plane slows downthe vertical part of the motion without changing the natureof the relationship between distance and the square of time.In other words, without changing the uniform acceleration.He reasoned that if freefall acceleration was constant,then so would the acceleration of the balldown the ramp be constant, but, at a slower rate.Which was related somehow to the slope or the steepness of the incline.This was assuming, of course, that the effectsof rolling resistance could be minimized.In other words, the incline changes the rate of the uniformacceleration without changing the uniformityof the distance/time relationship itself.

The details of the experiment will becomequite involved as we analyze them.We do this partly in respect for Galileo's genius, but mostlywe do it because it has become second nature to us in ourscience to conduct ourself in this manner with these experiments.It's so much of our paradigm that we do it more or lessunconsciously, when we do science.Sort of like riding a bicycle.We do these things automatically today in science, it's our paradigm.Like any cultural behavior it needs no justification,other than the fact that it works.If we're not careful, we're likely to forget that this paradigmarose out of formalized efforts like thoseof Galileo's, who had to come up it in the first place.Unlike us, he had to be aware of these details and think themthrough consciously in order to formulate the conceptof the experiment and to carry it out.You'll probably have the opportunity to do similarexperiments as part of your laboratory exercises for the course.

The design of an experiment is the most important part.In the same way that asking the right question is importantin getting the right answer, the results of an experiment can beno better than the design of the experiment.In other words, if you don't design a good experiment,then your results can't be any good.There's no recipe for designing an experiment.But Galileo's method left little room for improvement.Other than a statistical error analysis which wouldn't beavailable for another 200 years, there's little that we haveadded to the methods since Galileo did it first.

A hypothesis is a conjecture.It's a conjecture which is to be tested logically,mathematically or quantitatively.In an experiment it's a deductive statement about the resultsof the experiment to be expected if certain relationships are present.The best hypotheses are those which are simply statedwith few qualifying statements or special conditions.Galileo's hypothesis is actually composed of several parts.His hypothesis is that objects in freefall areuniformly accelerated in the absence of air resistance.So, one assumption here, is that the same rate of freefallapplies to objects regardless of the weightor gravity as Galileo called it.Another assumption is that the rate does not dependon the size, the shape or the composition of the material involved.In order to measure the rate of something, we must know what that thing is.We've seen how Galileo defined the terms in such a way thatthey depended upon measurable quantities.If you don't remember this, go back and reviewProgram 12, Lesson 2.3, if you need to.It might also help to review the laboratory exercise entitled,"Describing Motion," if you have received that exercise.We also saw in Lesson 2.3, the lesson on describing motion,how Galileo derived the relationships that could be usedto determine whether or not acceleration is uniform.

We know of two ways because there are two different relationships.One relationship is that the rate of change of velocitywith respect to time is constant.In other words, a graph of velocity versus time is linear.The second one which is more easy to test in an experiment is thatthe distance changes with the square of time, or the second power of time.In other words, the graph of distance versus timeto the second power is a straight line.A graph of "D" versus "T" squared is a linear graph.See, Galileo did not have the ability to measure velocitydirectly, nor did he have our ability to do graphical analysis,but he could make measurements and compare themwith the calculated values or compare the ratios.In Galileo's time there were no speedometers or accelerometers.He could measure average speed over long time intervals,but he could not show the details of the accelerated motion.What he could measure were distance and time.So he found it necessary to define uniformacceleration in terms of distance and time alone.This, as we have seen, involves algebraic logic, which wasknown to Galileo, although he didn't have the benefit of our modern notation.So here is that relationship again, the way Galileo derived it.Now this is really a simple substitution logic, so don't get lost in it.In words it would sound something like this, and you'll see herewhy the mathematics, I think, is so useful.

So first, we define average velocity as distance dividedby time so it must be true that distance is equalto velocity multiplied by time.Second, we define acceleration as the rate of change of velocity,or the change in velocity divided by time.If we start from rest, then the instantaneous velocity at anytime is the acceleration multiplied by the time.In uniformly accelerated motion, number 3, the average velocityis one half of the instantaneous velocity at the end of a giventime period, assuming an initial state of rest.So, combining all these, yields the relationship.Distance equals velocity multiplied by time, but averagevelocity is one half of the final instantaneous velocity.So, the distance must be equal to one half times the accelerationmultiplied by the time, multiplied by the time.Notice how the factor of time appears twice, multiplied by itself.So, then the distance traveled must be proportionalto the second power of time with a constant of proportion equalto one half of the acceleration, again,assuming initial velocity was zero.

So, the question is which one of these is easierto follow, the words or the mathematics?What makes it difficult to follow?What is it that makes it difficult?Is it the math, or is it the fact that we're discussing the rateof a rate, or is it something else entirely?OK now, the fact that the concept of time, which we don't reallyknow what it is anyway, could appear twice in the relationshipin this very algebraic way and yet still accurately describe the waysomething falls to the earth is simply amazing, if you think about it.Why should it be like this?Is this just Pythagorean, or do these relationships mean something?We may never know the reasons why these relationships exist.It may take forever if it can be done at all.But, see, that's not really important.Why the relationships exist does not affect the fact that it does exist.So, to understand how it works can be very simple.Even if we don't know why it works.So, the direct proportion between distance and time to the secondpower can be expressed in terms of a ratio.For a given rate of acceleration, doublingthe time results in quadrupling the distance.We saw this again in the last program.

Doubling the time results in quadrupling the distancefor any given distance and time.For example, if the ball rolls ten centimeters in two seconds,in four seconds, that's twice as long, it will roll fortycentimeters, that's four times as much.Twice the time, four times the distance.Four is two squared.Distance is proportional to the square or to the second power of time.In six seconds the ball will roll 90 centimeters.So how far will it roll in six seconds if the acceleration is constant?Well, six is three time two or three twos.The time is tripled so the distance will be ninetimes greater, because nine is three squared.Again, the distance is proportional to the square of time.So, measuring the distance and time as accurately as possiblewill make it easier to interpret the results and to see therelationships since you're looking at thisrelationship between distance and the square of time.Well, of the two, distance is very easy to measure and fairly easy to control.All you need is a stick or as in Galileo's case, he used agrooved board with small holes at even distances into which heinserted pegs to stop the rolling ball at a given distance,and also to give an audible clue as to when to stop the timer.The timer was the problem.

Galileo tried using a pendulum clock as a timer, but found itdifficult to measure the distance accuratelywhile counting the swings of the pendulum.You know you start the pendulum swinging and you roll the balland you have to get it like a fraction of a pendulum swing, it's not easy to do.So, to solve the problem of the clock he used,invented actually, what he called a water clock.Now this is a very ingenious thing too.If you have a big enough tank with a small hole in it,the water will flow out of that tank at uniform rate.You know this as well as I do that if you have a tank,a container with a hole in the bottom, the water will shootout of the bottom and if the, as the water gets lower and lowerin the tank, the rate of water slows down,So, big tank and a small hole over a short period of time the rateof water coming out is very constant.So, Galileo used this to his advantage.And what he would do, he would start and stop the watch,his water clock, simply by taking a container empty, putting itunder the water, when he wanted to start the clock, removing itwhen he wanted to stop the clock and simply by weighingthe amount of water in the container with a fairly accuratebalance, he had a measure of how much time had passed.Twice as much water in the container meant twice as much time had passed.Very, very ingenious.

Well, now we're ready to consider what Galileoactually discovered as a result of these studies.What I want to do is to show you and go through these veryquickly then we'll come back and look in more detailsome of the more important ones.So one of the things that he did was to show that accelerationdown the incline plane is actually uniform.That is, that the distance and the square of time are related aswe've seen in these, in the analysis, the mathematics that we had before.Also, the rate of accelerationon the incline depends upon the ratio of the height to length.And we'll come back to that one in a minute.The speed does not depend upon the weight or size of the ball.In other words, balls of different sizes and weights and made outof different materials accelerate down the rampat the same speed for a given slope.Thus disproving Aristotle's hypothesis that they fallaccording to their weights.Also Galileo discovered that the speed depends upon the resistance of the surface.That simply means that if you use a smooth ball on a smoothsurface, that it will roll faster on a given slopethan a rough ball on a rough surface.

We'll come back to this a little bit later on to see how Galileo usedthis actually to come up with the principle of inertia.Also, Galileo discovered that the speed of the ball at the bottomof the slope depends not on how steep the slope is, but the height.That is the height from which the ball is dropped.This is important enough that we'll come back to this one in a minute as well.He also then discovered that the uphill height equals the downhill height.That means that if you allow the ball to roll up a slopeon the other side, it will come to the same height that you rolledit from in the first place, and again, we'll come backto this one as well, because it's important.OK, now, what Galileo discovered was that the rateof acceleration on the incline depends upon the steepnessof the slope, but there's always a problemwith how you measure the steepness of the slope.The way Galileo decided to do it was to measure the ratioof the height to the length of the slope.As it turns out this particular relationship, the heightto the length, is what we now know in trigonometryas the sine of the angle of incline.If that means something to you, OK, if it doesn't, don't worry about it too much.This is kind of unexpected.Galileo discovered that the speed dependsupon the height and not on the slope.

Now, this is bizarre, if you think about it, because on one handyou look at this and you say, OK, the yellow ball on the righthere on ramp "B" is going to accelerate much faster thanthe ball on the left, the red ball on ramp "A".But keep in mind that speed depends upon not onlyacceleration, but also on time.Right?The velocity is acceleration multiplied by time.What Galileo discovered was that the ball on the right, the yellowball, will indeed accelerate faster.But because it travels a shorter distance downthe ramp, it will accelerate for a shorter time.So, its velocity is a large acceleration over a short time.The ball on the left, on the other hand, will accelerate slower.It's on less steep slope, so although it accelerates slower,it has a longer distance to travel, and so it willaccelerate for a longer time.So, let me review this for a minute.On one hand you have the yellow ball which will accelerateat a high rate for a short period of time.On the left hand side you have the red ball which will accelerateat a smaller rate, but for a longer period of time.And as it turns out, the two effects exactly cancel each other out.

Now we can show this mathematically if you wantedto take the time to do it, but I don't think it would really help us to do this.But the interesting thing is that the effects of the longerdistance and the shorter acceleration on ramp "A"exactly counter each other out so that the balls havethe same speed upon reaching the bottom.Now, you might understand here that we would know what speedthe balls have simply by taking the total distance down theramp and dividing it by the total time, which would give us theaverage velocity and we know that in uniform acceleration,the average velocity is exactly one half of the final velocity.So, if the two balls have the same average velocity, then we knowthat they also have the same final velocity.Oh, what is the highest level to which the ball can roll?If, for example, you design a ramp where the ball rolls down oneside and rolls up the other side, well, it's obvious that the ballcan roll no higher, I think it's obvious, anyway, that the ballcan roll no higher than from the height from whichit was rolled in the first place.

Galileo discovered this somewhat accidently,but he also put forth a hypothesis that went something like this.He said, "I don't know what it is that makes the ballaccelerate going down the hill.I don't have to know what makes it accelerate.All I have to know is how it accelerates and what speed it reaches.""What I do know," he said, "is that whatever causes the ballto accelerate down the ramp on one side also causes the ballto decelerate going up the ramp on the other side."So, if things reach the same amount of speed from a givenheight, it seems logical that if a ball gains a certain amountof speed rolling from a certain height, that it will also loseexactly that amount of speed rolling uphill to that same height.And sure enough when he tested it, it turned out to be the case.And you can see on the picture on the screen thatthe ball rolls to the same height.

Now, I have to point out here that Galileo went much further thanthis and also noticed that the ball will never actually reach that height.In other words, the surface and the roughness detractfrom the motion and they take away some of the motion of the ball.But, the smoother he made the surface and the smoother hemade the ball, the closer the ball came to rolling back to that original height.In fact, on a very, very smooth surface, a groove with the ballrolling on it so it only touches on the sides of the groove, you willfind that the ball rolls almost exactly at the same height.I think if you consider this for a minute you should be able to seesome relationship here between the ball rolling down one hilland up another and the pendulum.What will happen, for example, if you simply let the ball roll backand forth, up one hill, down the other, then down that hill,up the other, you'll see that the ball has a motionthat's not too different from a pendulum.So from these experiments Galileo drew three major conclusionswhich would lead him to his explanation for projectilemotion replacing Aristotle's weakly justified notionof antiperistalsis and also combining the conceptsof violent and natural motion into a common set of rules and terms.Not only that, but it would also put into place a new paradigmand stimulate a whole new era of mathematical analysis whichIsaac Newton would nail down a generation later.

So, the first of these conclusions, we've already seen is that theinteraction with air and surfaces interferes with the purity of motion.The second one we've also already seen is that freefall is uniform acceleration.We'll come back in a minute and see a picture of this.And the third one is that inertia causes matterto remain in motion or at rest.But we haven't encountered this term, inertia, yet so we'll takea minute to come back to this and look at it in a little more detail.Here's a picture, a strobe photograph taken of a falling weight.Had Galileo had the ability to take a strobe photograph like this,I don't think he would have had to spend as many hours as he did.What you see here is a photograph which is taken in the darkby a strobe light which flashes at a regular interval.In this case it's 1/130 of a second.So, what you see here is that the weight actually fallsincreasing distance at each time interval.It's very difficult to see from just a picture that it's uniformlyaccelerated, but you can actually measure the rateof acceleration in freefall this way, if you have a scalein the background and you can simply measure how far theprojectile or in this case, the weight has fallen in each 1/30 of a second.It's the concept of inertia that we really want to focus onin this particular analysis and conclusions,because here's what's happened now.

Galileo has recognized that the ball will roll to the same heighton the uphill side as it will rolled from on the downhill side.So, here's what he's recognizing now that there are two thingsthat interfere with the motion of the ball.First of all, remember that, something causes itto move down the hill in the first place.That same something causes it to slow down going up the hill on the other side.OK?"So, what things," he asks, "slow down the ball going up the hill?"Well, one thing is gravity, it's the weight of the ball.But the other thing is friction.Right?It's the interaction of the friction with the air.So the logical question that Galileo asks is suppose youdecrease the slope of the uphill ramp and allow the ball to rolla greater and greater distance before it reaches that height.

Now, in the absence of friction, the distancethat the ball rolls makes no difference.And so, and this is the real genius in Galileo's analysis here,how far will the ball roll if there's no friction and if it's not rolling uphill.In other words, rolling on the level.Galileo's answer to that, you actually can come to only oneconclusion, that if there's nothing to stop the ballfrom rolling, that it won't stop rolling.If there's no gravity in the form of a hill, if there's no frictionin the form of rolling resistance, then the ball will keep on rolling.This is entirely contrary to Aristotle's view of motion.It's also contrary to our senses, but as we'll see later,it removes the idea that necessarily puts the prime mover in the heavens.And it will open a whole new paradigm.It will open a whole new door to look at motion in an entirelydifferent way once Galileo has come to this conclusion.So what we have here is a whole new paradigmregarding the nature of motion.Galileo's discovered that the natural laws apply to everything.The same laws apply to all kinds of matter.Governed by mathematical relationships and that thephysical behavior of nature is such that we're interestedin how it behaves, not what causes it to behave that way.And finally, it's the nature of motion itself that Galileohas come up with a new paradigm here.

The idea that motion is a natural state which continues until altered.Replaces Aristotle's old idea that motion is anunnatural state which ceases unless maintained.You probably want to stop and think about these twostatements for a minute to make sure that youunderstand what the difference is between them.So at last we're ready to see now what Galileodiscovered with the inclined plane and freefall hasto do with projectiles and how it explains Aristotle's,I should say refutes Aristotle's view of projectile motion.What I want to do is to take you to look at some animations thatwe've done and we'll talk about those as the animations come up.

OK.Here's an animation of a ball rolling off a flat surfaceand launching itself as a projectile.The first thing we notice here is that it definitely follows a curved arc.Remember, Aristotle had said that things onlymoved in curved arcs in the heavens.So what we'd like to do now is to take this motion and analyzeit in terms of what Galileo discovered with the inclined plane and freefall.So, keeping in mind, now, that Galileo haddiscovered that there were only two things that slowdown the motion of something that's moving.One of those is resistance to the air or rolling, the other is gravity.So let's suppose we could shut off both of thesethings and throw a ball horizontally.So, we would discover that the ball moves along at a constantspeed moving inertially; that is, there's nothing acting to slowit down, and so it doesn't slow down.So it moves equal distances in equal intervals of time.The graphic on the screen shows this.We've stopped the ball at each time as if we were taking astrobe photograph of it, and you'll see that it definitely does coverthe same distance in the same amount of time.

So now let's turn gravity back on, but still leaveout the effect of air resistance.As we saw in the strobe photograph before an objectin freefall is accelerating and so it covers a greaterdistance in each successive time interval.The animation on the screen shows this.This is a ball being dropped.It's little slowed down, but it's still, you see that each flashor each time interval, it covers a greater distance.What Galileo recognized is that the rate of freefall is constantfor all objects regardless of their size, weightor composition in the absence of air resistance.So, he asked himself why should the rate of freefall be anydifferent if the object happens to be movingforward at the same time it's falling?In other words he asked the question, "Will the ball takethe same time to fall from a certain height even if ithappens to be moving forward?"When he came up with the answer to that,and logically it seems that it would not.I mean if you think about it this way.If you're flying in an airplane at 600 miles per hour and youdrop something, what happens to it.Well, of course, from your perspective inside the airplane,it appears to fall straight down, exactly as it doesif the airplane was sitting still on the ground.

So from your perspective, the ball falls straight down, but whatdoes it look like to someone outside the airplane.What would it look like?Well, obviously the amount of time it takes the ball to fall thedistance to the floor, the airplane has moved several hundred feetbecause it's traveling at 500 miles per hour.So what would the path of the ball look like?I think if we look at the next animation, we can put this alltogether and see now how projectile motion can becomposed of both freefall motion and inertial motion.So now if we look at the combined motions, the inertial motionin the horizontal direction and the freefall accelerated motionin the vertical direction, we see that we can exactly recreatethe original path of the projectile.You'll notice that the ball that's falling in the parabolic arc isalways exactly at the same height as the ball that's falling in freefall.And at the same time, it's always directlyunder the ball that's moving horizontally.What's happened here is that Galileo has recognized that thefreefall motion is combined with the horizontal motion so thatthe projectile actually is doing two things at once.Right?It's moving at constant speed in the horizontal direction.At the same time it's accelerating in the downward direction.

Now this is a great revelation, and it's also very parsimonious.Because what he's done here is to take something that appearsto be very complicated, the motion of the projectile,and break it into two sets of simpler motion and simplylook at each kinds of motion separately.It's very parsimonious.It's also a stroke of genius.Galileo was very quick to recognize that if a ball isthrown vertically upward, it will lose speed against its gravity.In other words, it will slow down as it moves upwardand eventually come to rest, instantaneously, come to restat the top of its arc and then turn around and begin to gain speed again as it falls.The upward and downward part of the motion are perfectly symmetrical.So, the same is true, even if the ball is thrown upward whileit's moving forward, for example, inside the car, the bus or the airplane.The situation is exactly analogous to launching the projectile at anangle, like hitting a golf ball with a golf club or kicking a football.The only variables involved here are the speed of an angle of the launch.

The rate of freefall acceleration does not change just becausesomething is moving horizontally.In fact, the time for it to go up and come back down againis the same whether it happens to be moving forward or not.The strobe photograph you see on the screen shows the actualpath of a projectile, and you can't help but notice a symmetry involved here.In fact, the symmetry is such that if you look at the right handside of this parabola, you'll notice that it looks exactly the sameas the projectile that was launched horizontally off of a building.I use the word parabola here.Is that coincidental that this is a parabola?A parabola is, after all, one of the conic sections, isn't it?So in this animation of a projectile cart, you can seethat when the ball is launched vertically upwards, its upand down motion is identical to that of a ball thrownvertically by a stationary observer.While the horizontal motion is exactly the same as the cart.So this is what would happen if you were riding in the bus or anairplane or a car and threw something up to catch itlike you were juggling, for example.It would still land exactly in your hand, the same as it wouldif you were standing still, assuming that the vehicleyou're riding in doesn't accelerate.

In the next lesson we'll consider what might happenif the vehicle accelerates, but it's not too early for you to start thinking about it.Well, Galileo's list of accomplishments really is impressive.Not only did he do all these things, but he also formalizedthe scientific method and set a new paradigmfor learning about the physical world.Although he was considered a criminal in his native Italy,Galileo's work made available through his writings, had greatinfluence in Europe and England where the social climate wasa little, well actually, a lot, less rigid.So, with Galileo we see the establishment of a new paradigm for physics.This paradigm sets the stage for the rapid growthof experimental science, the use of mathematical analysis,the search for mathematical relationshipsand the understanding of the planetary motions.We'll take up Newton and the theory of gravityin the next two lessons to understand this.

Well, I guess that's it for Program 14 in Galileo.We'll see you next time.And remember, when it comes to science, get physical.So, what do you think of Galileo and his science my silicon friend?Silico: "I do not understand inertia.Maybe it is because I cannot feel these light forces and such."Music