In this lesson we will learn about Aristotle's views on motion as incorporated into the Scholastic Philosophy, and we will learn how we describe motion in moderns terms. We will learn the definitions of speed, velocity, average velocity, instantaneous velocity, and acceleration, and the relationships between them. We will see how motion is described in terms of distance and time, and we will begin to learn about the use of graphs and geometric figures in studying physical phenomena such as motion.
In this lesson we will contrast the ancient and modern views on motion in preparation for understanding Galileo's experiments. He did those experiments because he recognized that the Scholastic views on motion, inherited as they were directly from Aristotle, we not correct. Our modern description of motion is entirely due to Galileo, who defined the terms and derived measurable relationships from those definitions.
The first section, "Aristotle & Scholastic Physics" is a summary of the views of motion which were current in Galileo's time.
The second section is a modern description forged out of Galileo's definitions of velocity and acceleration in terms of distance and time, and Descartes' marriage of algebra and geometry into a numerical graphic tool which was indispensable in Newton's formulation of gravity, and to all subsequent physical science.
It is through the understanding of motion in general that we come to specifically understand the motion of planets, interplanetary rockets and baseballs.
Piaget conducted experiments which indicate that this concept of motion is that of the child. They represent the first approximation of a description of motion based largely on intuition.
Aristotle formalized the study and argued logically, but he lacked a quantified definition of the terms used to describe motion.
We have already noted that of all of Aristotle's wonderfully complex and well thought out world views, his theories of motion were the weakest and contained the most holes.
It was as if Aristotle painted himself into a logical corner, then tiptoed out through the wet paint leaving footprints that he hoped couldn't be seen.
As we learn about Aristotle's concepts of motion, try to relate them to your own experiences with motion. Do you agree that they seem reasonable according to your own perceptions of motion?
Consider that in Aristotle's time the fastest a person could travel was on a galloping horse or in a chariot pulled behind one. It's very hard to objectively observe the effects of motion while bouncing around in a chariot or on horseback.
In Aristotle's time the smoothest motion was on a ship at sea under sail. Here the effects of wind and the relative slow rates of speed make it difficult to study motion.
What Aristotle and his followers needed was a faster car with good suspension and a smooth concrete ribbon to go gliding a mile a minute on wheels made of air and rubber.
Or else someone had come up with a way to study motion objectively and repeatedly under controlled circumstances.
Galileo did this, but only after he had defined the parameters of motion and their relationship in simple terms.
As we will see in future lessons, once we know what the parameters are, then, and only then, can we figure out a way to measure them, then we can begin to do experiments like Galileo did.
You may wish to review lesson 9 to refresh your memory about Scholastic philosophy and its views on motion.
The Scholastic philosophy arose out of the need to study the ancient documents. The main ideas underlying the Scholastics was a synthesis of Aristotle's cosmology, his views on nature and motion, with the mathematical philosophy of Plato including it's Pythagorean mysticism and circular perfection. Added to that was the geocentrism of Ptolemy where the epicycles represented actual motions and not just mathematical convenience. All of this was combined with Christian doctrine and dogma to produce a strong system of knowledge and learning.
The Scholastic method became the standard paradigm for studying and for for acquiring new knowledge. It is difficult to express precisely how such a paradigm works, but one of the pressing questions had to do with the size of angels.
The question of how many angels could fit on the head of a pin was a topic of debate and serious scholarly research, as was the size, shape, and precise location of hell as described as "The Inferno" in Dante's Divine Comedy.
A typical argument is illustrated by this quote from a prominent astronomer of Florence who reacted with authoritative disdain to Galileo's report of his own discovery of the four largest moons of Jupiter.
"There are seven windows in the head, two nostrils, two ears, two eyes and a mouth; so in the heavens there are two favorable stars, two unpropitious, two luminaries, and Mercury alone undecided and indifferent. From which and many other similar phenomena of nature such as the seven metals, etc. we gather that the number of planets is necessarily seven. Besides . . . we have the division of the week into seven days named for the seven planets: now if we increase the number of planets, this whole system falls to the ground . . . Moreover, the satellites are invisible to the naked eye and therefore can have no influence on the earth and therefore would be useless and therefore do not exist."
Francesco Sizi, Florentine Astronomer responding to Galileo's discovery of Jupiter's moons.
From this critique we get a feeling for the type of logic, and the type of questions which might be considered appropriate. Notice specifically what kind of things are taken as given.
For example, the significance of the number seven which corresponds to the number of metals, planets, and days in the week. These things are not all in the same category. Certainly the seven day week is a construction of the human mind, having been passed down from the Babylonian calendar.
Also note how this strict adherence to the numerical significance precludes discovering more planets or more metals, the paradigm insisting that the number is fixed at seven.
In the Scholastic world view, the universe is geocentric and circular, but its motion is part of God's plan for salvation. Incorporating celestial motion into the divine plan made it even more difficult to view the motions of the heavens in the same terms as earthly motion.
The deferents and epicycles of Ptolemy represented real motions and not just mathematical convenience. You may recall that Ptolemy had dismissed any links between the planetary movements, whereas Aristotle insisted on unity and connections.
Today our use of mathematics is focused on describing and predicting changes. The calculus, an advanced version of analytic geometry, was invented independently by Newton in England and Liebnitz in Germany specifically to describe how one variable changes in relation to another.
Aristotle argued that change was too ethereal to be described by mathematics. Mathematics was useful in arithmetic for adding and subtracting, and in geometry for calculating angles, perimeters, and areas, but certainly could not describe or predict change.
2.8.1. useful for calculations but not for descriptions
In Aristotle's cosmology, the matter dominates the universe and therefore it is the properties of matter which drive all types of change and determine the outcome of those changes.
2.9.1. prime substances and qualities = elements
Aristotle's scheme for matter allowed for the possibility of one substance changing into another by addition or removal of the elemental qualities.
220.127.116.11. earth, air, fire, water
18.104.22.168. hot, cold, wet, dry
22.214.171.124. elemental qualities responsible for motion and chemical change
In this scheme all motion and chemical change was due to the desire for the four elements to purify themselves and separate into the concentric layers.
126.96.36.199. relation to chemistry later
In later lessons we will learn how this view influenced the development of alchemy and chemistry. Aristotle's failure to distinguish between physical and chemical change may seem shortsighted, but we will see that such changes are not always easily distinguishable.
2.9.2. Properties of matter are related to location in universe
In the concentrically tiered model of the universe which Aristotle so aptly described, the Heavens are composed of an imponderable substance, called quintessence. It is different from earthly matter, but since it is imponderable we will never know exactly what it is like and anyway it is beyond our ability to understand so we shouldn't think about it. Because there is perfection in the heavens, there can be no change, except for the heavenly motions, which he did not really view as change in the same way that changes take place on earth. The heavenly motions were part of the heavenly perfection and so by definition were not changes, but a type of constancy (hence the necessity for uniform circular motion.)
In the sublunar realm, chaos and imperfection reign. The pure elements, earth, air, fire, and water, find themselves combined into the variety of substances in the physical world and they would like to be separated. A good analogy would be a shaken bottle of oil and vinegar salad dressing which, if left undisturbed, will eventually settle into layers.
Aristotle explained that all change in the sublunar realm
188.8.131.52. Heavens are perfect quintessence
184.108.40.206.1. heavenly motion is pure and perfect
220.127.116.11. Earth is imperfect mixture of prime substances
18.104.22.168.1. sublunar motion is unnatural because of imperfection
22.214.171.124.2. changes occur to attain purity and perfection
126.96.36.199.3. prime substances are mixed, want to be unmixed
Aristotle believed that motion itself had certain properties, which he claimed were intuitively obvious. As we will see, this is one good example of a case where so called common sense leads us far astray.
These properties, incorrectly assumed as given, were used all the way into the seventeenth century to "prove" that the earth could not be moving around the sun or rotating daily on its axis.
Obviously, Aristotle said, rest is the natural state of the universe because most things we see are not moving. To get them to move we have to actively move them and when we stop moving them, they stop moving. For the heavenly motions, this belief necessitated the Prime Mover to turn the crank and keep the celestial sphere and its heavenly bodies turning.
Observations suggest to us that heavenly motion is circular, but motion here on earth is linear. After all, when a ball rolls down a hill, it rolls in a straight line, or at least it would if the surface did not disrupt it. Similarly, an object in freefall falls downward in a straight line.
188.8.131.52. motion will cease when cause is removed
184.108.40.206. heavenly motion is circular, sublunar motion is linear
Once the cause of motion is removed then motion stops. Try it. Push a pencil across a tabletop, then let go of it. It will very quickly come to rest. You are the cause of it's motion by acting upon it and when you stop, so does it.
As for natural motion like freefall, or pebbles falling through water, Aristotle had claimed that this was due to the natural and preferred positions of the various prime substance in the heavenly hierarchy.
A rock fell through air and water because its composition is more earthly that the other elements so its natural place is in the bottom layer. Sure enough, a rock will fall through fire, air, and water, and come to rest on the bottom with the other earthly substances.
Similarly, fire rises through air and air rises through water to because that is where it belongs.
Aristotle also thought that water flowing in streams was issuing from underground springs as the water sought its natural place in the ocean.
220.127.116.11. ultimate cause of all heavenly motion is Prime Mover
18.104.22.168. ultimate cause of unnatural sublunar motion is necessity for perfection and purity
Aristotle also gave several arguments against a moving earth based logically on these perceived properties. These arguments were used throughout the middle ages whenever the subject of a moving earth was brought up. The logic is similar to that of astronomer Sizi, quoted above.
22.214.171.124. spinning Earth would drag air
Since earth would be spinning under the air the motion would drag the air causing tremendous wind, similar to the way the wind blows when you stick your hand out the car window. Of course, Aristotle had no car, but the effect is noticeable even at low speeds, so at the tremendous speed required to turn the earth in 24 hours the wind would be unbearable.
126.96.36.199. nothing to keep it spinning
Since, according to Aristotle, action is required to sustain motion, there is no obvious cause which keeps the earth spinning. It would naturally stop spinning in the absence of such a force.
It's not clear why he could not have invented an imponderable mover of some sort to do the job, but he didn't.
188.8.131.52. objects thrown upwards or dropped would not fall vertically
An object thrown upwards would land at the same spot it was thrown, but if the earth moved out from under it then it would appear to land behind the spot. In this condition, you might jump into the air and be hit from behind by a tree, a building, or another person.
This doesn't happen, so obviously the earth can't be moving.
184.108.40.206. loose objects would be thrown off the surface
Like water being flung from a wet shirt whirled overhead, objects on the earth would be thrown out into space if the earth was rotating.
This one seems rather natural, especially considering that Aristotle had no concept of gravity as a force which holds us to the earth.
Aristotle described four distinct and separate kinds of motion, unrelated, having different causes, and following different principles. We will see the unification of these different types of motion beginning with Galileo and culminating with Newton's gravity.
The type of motion that Aristotle called alteration referred to what we would call chemical change today. We might even have to stretch the imagination to call it motion at all, although we certainly recognize it as a type of change.
Aristotle's description of vertical motion was based partly on his observations of the way in which different objects fell through water. He tried to generalize from water to air by correctly assuming that air was a thinner version of water and would affect motion in a similar way. He was incorrect, however, in that he observed objects falling through water at their terminal velocity because water is really a lot thicker than air. We might say he was generally confused about the relationship between falling through air and through water.
220.127.116.11. Aristotle's Observations
Aristotle actually observed the way in which objects of various sizes and weights settled through water. He correctly induced that weight does affect the motion, but he failed to appreciate the short time required to reach terminal velocity, and the behavior while reaching that speed.
18.104.22.168.1. watched things settling through water
22.214.171.124.2. correctly induced there is some effect of weight under these conditions
126.96.36.199.3. failed to appreciate the short time to reach final velocity
188.8.131.52. Aristotle's Explanation
He decided that there could be no vacuum since the medium through which an object is falling controls its speed, and that water slows an object more than air. In the absence of any disturbing Aristotle tried to explain his observations in terms of the desire of a material to be in its natural place. Deciding that earth is more waterlike than airlike, it seemed obvious that earth would fall faster through air than water.
Since heavy objects contain more matter, they would be in more of a hurry to find their place and so would fall faster. He correctly observed that heavy objects fall faster in water but incorrectly generalized to air. He stated that even in air heavier objects should fall faster, and at a rate proportional to their weights. In quantitative terms he was stating that if one objects was twice as heavy then it would fall twice as fast.
In the absence of a medium such as water or air there would be nothing to control the speed at all and it would therefore be infinite. Since the concept of infinite speed doesn't make sense then there can be no vacuum. Thus the statement, popular well into the later centuries, "nature abhors and vacuum."
184.108.40.206.1. movement is up or down according to natural place
220.127.116.11.2. movement is slower in dense materials like water
18.104.22.168.3. light and heavy objects fall at different speeds controlled by medium
22.214.171.124.4. infinite speed in vacuum so there can be no vacuum ("nature abhors a vacuum")
2.11.3. violent motion: projectiles, things pushed or pulled
Violent refers to the action necessary to move things horizontally. Unlike natural motion which happens spontaneously, violent motion does not happen without action. The idea that something must continually push objects clouded the explanation. Today we recognize that friction with the air or where surfaces are in contact, is the cause of the loss of motion, and it is friction that must be overcome to keep an object moving.
Explaining the movement of projectiles, objects that were thrown, was the most difficult. This is one of those places where he painted himself into an intellectual corner. The more he considered, the deeper into the corner he got. He never did really explain his way out of this one.
To explain projectiles he imagined a process that he called antiperistasis. Any object traveling through the air must push air out of the way. Antipersistasis is the force of the air rushing back into fill the vacuum left as the object passes through. Since nature abhors a vacuum the air will forcibly strike the projectile and continually propel it through the air.
As you might suspect, this was the weakest of all of Aristotle's theories. It did not explain, for example, why the projectile followed a curved path. In fact Aristotle stated that it did not, that it moved in a straight line until it stopped, then fell straight down. It is obvious that this is not what happens if you simply throw a ball from one hand to the other. It also does not explain why the projectile ever loses speed at all.
We have already seen in lesson 9, that there was significant criticism of Aristotle's theories of motion during the middle ages and into the reformation. As we will see, Galileo recognized these weaknesses as a good place to attack Aristotle's authority.
126.96.36.199. notion that something must continually push objects clouded explanation
188.8.131.52. antiperistasis: air rushes in behind to push
184.108.40.206. weakest of all motion theories
2.11.4. celestial motion
We have already seen that celestial motion is a special kind of motion since the heavenly objects are imponderable and massless quintessence which move somehow unimpeded through ethereal crystal spheres driven by an equally imponderable Prime Mover.
220.127.116.11. heavenly objects are massless quintessence
18.104.22.168. move unimpeded through crystal spheres
22.214.171.124. driven by imponderable Prime Mover
3.1.1. how is it similar?
3.1.2. how is it different?
Now we can look at our modern view of motion. We will use a form of symbolic language which is commonly referred to as mathematics, although the word itself is enough to cause fear in many people.
Do not be alarmed! The function of these symbols is to represent relationships between the parameters of motion.
At this point you might want to make a note to look up the word "parameter" in the dictionary. After you have done that, continue reading.
Here we would like you to think of a parameter as a particular state of a system. For example, a ball will roll downhill on a flat board, but the angle or slope of the board will affect the motion. So we would say that the slope of a hill is a parameter which we might want to measure, but also to control and vary systematically.
For example, to discover quantitative relationships between parameters we could measure the distance traveled in a certain time and compare it to the slope. Do you see what a parameter is now?
Make a short list of some parameters which might affect a daily routine of some kind, like driving to work or to the store.
In this lesson we will establish a relationship between the parameters of motion. We will start with Galileo's definitions and end with a more modern graphical description as we learn a new and amazing connection between numbers and shapes, undreamed of even by Pythagoras himself, and the key to understanding the motion of the planets . . .
Be sure to study the examples in the text, in the chapter which covers motion. If you made the table of contents as we suggested earlier, it should be easy to find it now. If not you might want to do that now.
Do not be put off by the equations and the word problems in the text. They are expressing relationships, and the numerical problems are a way to help conceptualize them. Look at the examples to see how the problems are solved. Try one or two of them yourself. If you can solve the problems it is good. If you cannot solve them it is OK. Do not get discouraged and give up if the equations do not register with you.
It is the relationships and not the equations that are important. To truly understand motion, you must understand the relationships. The equations should serve only to organize the relationships.
In the lab exercises you will explore the nature of linear and non linear relationships in a quantitative way.
As you begin to discover these and understand them you will see that there are many different ways to organize our intuitions, and the mathematical notation is one way.
Regardless of how well you relate to the mathematical symbols, you should try restating the equations in words and try to conceptualize the meaning. This will improve your mathematical sophistication and also help you in upcoming lessons.
Our modern understanding of motion is very different from that of Aristotle and his Scholastic followers. For many of us Aristotle's description seems to follow common sense. It does. Unfortunately, common sense doesn't always give us the right answers, and with motion we have to look a things a little differently.
This is what is necessary to comprehend the physical world. It takes a different way of thinking, the use of precise definitions, establishing mathematical relationships, and using models as a way of problem solving.
In this section we will define the basic parameters of motion in terms of distance and time, and establish relationships between these parameters which will allow us to think clearly about motion.
4.1.1. describing reality is goal of physical science
Like the ancients our goal in physical science is to describe reality. We recognize that there may be a physical reality distinct from spiritual reality. We have developed a preference for the quantitative over the qualitative, especially when it comes to deciding on the truth of our conjectures. We rely on numbers in the form of data which we collect from measurements. We look at relationships and types of relationships between numbers and shapes, and we translate between numbers and shapes as models for testing our understanding against reality.
4.1.2. mathematics allows us to combine all three to simplify understanding
The techniques of mathematics which have evolved since Aristotle's time have grown immensely in sophistication. We now know how to see relationships between numbers in terms of graphical shapes, a remarkable discovery made by Rene Descartes in the middle of the seventeenth century.
4.1.3. motion is natural state of universe
Today we believe that motion is the natural state of the universe rather than an unnatural state whose existence must be explained. In fact, everything is constantly in motion in our modern view.
126.96.36.199. only an illusion that things are still
The atoms that comprise matter of all kinds are constantly moving with thermal energy. In chemical reactions atoms are rearranged. The constant motion of the earth, the sun, the moon, the stars, the oceans, the wind, clouds, etc. are all part of a larger process involving the concept of energy.
We do not directly observe the persistent motion of atoms because they are so small. In a similar way, we do not observe the waves on the ocean from a jet airliner or from near earth orbit in the space shuttle. Relatively speaking, the atoms in a flat, apparently motionless tabletop are much smaller than the waves on the ocean from 35,000 feet.
188.8.131.52.1. surface of ocean appears flat and motionless from 35,000 ft
184.108.40.206.2. size of atoms in relation to flat tabletop is smaller than waves to flat ocean
220.127.116.11. set stage to understand motion of atoms in same terms as motion of planets
The understanding of motion which Galileo's insight and experimental genius gave us allowed us to eventually see that all types of motion, alteration, natural, violent, and celestial, can be viewed in the same terms. We now believe that the laws of motion apply the same to all objects regardless of size, even in apparently solid objects as well as planets.
18.104.22.168.1. Newton's Gravitation implied that laws of nature are universal, apply to all objects no matter how small
22.214.171.124.2. later scientists thought it should relate to atoms in apparently solid objects as well as planets
126.96.36.199. understanding motion allows relationship with forces to be seen
The relationship between motion and forces, as stated by Newton, was the breakthrough which finally allowed us to make the connections between planetary motion and atomic motion.
We will continue to explore these topics throughout the remainder of the course.
Now we are ready to get down to business with the concept of motion. First of all we will define what we mean by speed, then distinguish between speed and velocity, and between average and instantaneous speed and velocity.
It is useful, when trying to understand something, to first define what the thing is.
In an earlier lesson (lesson 2) we noted that it is impossible to study something until you know that it is a thing. The more specific the definition the better the odds that we know what we are talking about.
4.2.1. speed: distance divided by time
It is easy enough to decide which horse moves faster in a race. The one which reaches the finish line first must have run faster. That's intuitive.
But how would you compare the speeds of two horses who ran the same track on different days, or on different tracks on the same day?
Another way to think of the speed of the horses, is that the one which runs the fastest will cover the given distance (the length of the race) in the least time.
We will define speed simply as distance divided by time. Thus the horse which runs a given distance in the least time will have a higher speed.
Please note that in a fraction, the bigger the denominator (the downstairs portion) the larger the fraction. So 1/4 is a smaller number than 1/3.
So if one horse runs 1 mile in 120 seconds it has a higher speed than one who runs the same 1 mile in 130 seconds. The faster horse will arrive at the finish line ten seconds before the slower, and 1/120 is a bigger number than 1/130.
Although it seems obvious that this relationship adequately describes speed, the concept of time was never formalized in the description of motion until Galileo gave this definition late in the sixteenth century. This is partly due to the absence of accurate timekeeping devices, but also due to the changing ideas about the nature of time.
We will return to this topic sporadically in upcoming lessons.
So, if an object moves thirty feet in one second, we say its speed was 30 feet divided by 1 second (30 ft / 1 sec) or thirty feet per second. If it traveled 60 feet in 2 seconds, we say its speed was 60 feet / 2 seconds, or thirty feet per second. So we can compare the speed of different objects over different distances or times by this ration which we call speed.
Qualitatively we might think of speed as the rate of change of location or position.
4.2.2. average speed and average velocity
In some cases the direction of motion is also important. Presumably in a horse race all the horses are running in the same direction around the track, so we do not need to take that direction into account.
We can then define velocity as change in position divided by time, or the rate of change of location.
We have no guarantee that the horse did not speed up or slow down during the race. In fact, without more careful measurements, all we know is that the horse traveled a certain distance, in a certain direction, in a certain amount of time. We have no information about how that was achieved.
Recall the allegory of the tortoise and the hare. The turtle's slow steady pace compared with the rabbits erratic and changing pace allowed both of them to cover the same distance in approximately the same amount of time.
Similarly, when you drive from home to work or school, it is unlikely that you move at a constant speed in traffic. Your speedometer will show many different speeds. But regardless of the details of stopping and starting, speeding and slowing, if it takes you one hour to travel twenty miles, then your average speed was twenty miles per hour.
To be more precise we might specify not only distance, but also direction. Obviously two horse running in different directions at the same speed would reach different locations, so both speed and direction are important in describing motion.
We might say at this point that velocity is speed in a certain direction.
The distinction between speed and velocity is not important most of the time, and it is not important when trying to visualize the concept of speed as the rate of change of position or location.
When direction is important, such as in trying to figure out where you will be one hour from now if you average sixty miles per hour, we will use it. When direction is not important, such as when everything is moving in the same direction all the time, we will not use it.
Don't be confused by this. It is just a way to simplify our considerations by eliminating those things which are not important to understanding a situation, a step in the process of parsimony.
188.8.131.52. average is total distance divided by total time
184.108.40.206. velocity is speed in a certain direction
4.2.3. instantaneous velocity
What about the reading on the car's speedometer? Is that measuring average speed or is it measuring something else? What it is really measuring is instantaneous velocity.
220.127.116.11. average velocity over a small interval of time
We will define instantaneous velocity as the average velocity at a particular instant of time or over a small interval of time.
It is necessary to do this because we rarely move at a constant speed for sustained periods of time, for example during the daily commute. The speedometer on the car is actually measuring instantaneous speed.
The concept of instantaneous velocity is a little more abstract than average velocity. How are we to determine what interval is small enough to be considered instantaneous? We take some small time interval over which the speed does not change. No matter what the state of motion, there will be some interval where the speed is nearly constant. The limit of that interval is the instantaneous velocity.
As a qualitative example, suppose you look at the speedometer on a car which is braking as it slows from 30 mph (miles per hour) to 10 miles per hour. Instantaneous speed is the position of the speedometer needle at a given instant. But how long is an instant? Instantaneous speed could be measured by taking a photograph of the speedometer needle as it drops from 30 to 10. But a photograph isn't really instantaneous because the light must fall on the film for some period of time in order to register an image.
Normally we take photographs of things which are stationary, unless we want to indicate motion, in which case the moving object is blurred. Whether or not the needle is blurry or sharp on the photograph depends on two things, the rate that the needle is moving and the length of time that the shutter is open on the camera. The faster the speed changes, the faster the shutter speed (the smaller the time interval) that is necessary in order to show a sharp image on the photo.
To get a qualitative example, let's look at some numbers.
18.104.22.168. numerical example
Suppose a car is traveling at sixty miles per hour. Then in one hour, if its speed does not change, if will have traveled sixty miles.
At that rate the car will cover eighty eight feet in one second. You can check it out by doing a conversion if you want, that 60 mi/hr = 88 feet/sec.
It is much more likely that the car will move at a constant speed for one second and actually cover 88 feet in that time. If that is the case we could say that its average velocity over that one second interval is 88 ft/sec and we could calculate that if it continued at that rate for one hour it would travel sixty miles.
But suppose that during that one second the car speeds up or slows down?
Easy, then we could look at the distance traveled in a smaller interval, say 1/10 of a second. At 60 miles per hour, or 88 ft/sec, the car should travel 8.8 feet in 1/10 sec. But it would also travel 0.88 ft in 1/100 of a second and 0.088 feet in 1/1000 of a second and 0.0088 feet in 1/10,000 of a second, etc.
Eventually we will find some small interval of time during which the speed will not change measurably, then we have found the instantaneous velocity.
All of these ratios are numerically equivalent:
60 miles/1 hour = 88 ft/1 sec = 8.8 ft/0.1 sec = 0.88 ft/0.01 sec, etc.
They all represent the same speed even though the car may not continue at that speed except during the interval over which we measure it.
22.214.171.124.1. if traveling at constant instantaneous velocity will cover a certain distance in a certain time
126.96.36.199.2. constant velocity means that instantaneous velocity equals average velocity for long period of time
4.2.4. Summary of Speed and Velocity
If these concepts are not immediately clear to you then you will understand why Aristotle and everyone else until Galileo had so much difficulty with describing motion.
These concepts are clearly defined, and not really as difficult to understand as they seem. You will find it useful to separate the definitions and the meaning of the terms. Remember, one of the useful discoveries of modern times is that we do not have to completely understand everything in order to use it.
Like the VCR, we don't have to know everything about how motion works, but we do need to know where the controls are and what happens when we change them.
You should read the descriptions of motion in the textbooks and in the laboratory exercise on describing motion. Like anything new it takes awhile to assimilate and begin to feel comfortable with this concept of motion.
So, let us say this concisely. Instantaneous velocity is the average velocity over a small interval during which speed remains constant. Whenever speed remains constant then average velocity and instantaneous velocity are the same. An object which continues to move at a constant velocity will cover equal amounts of distance in equal times, and the distance covered in any time interval will be proportional to the time interval.
Can you think of other ways to state this relationship?
Now we are ready to deal with situations in which the instantaneous speed changes. To keep it simple, we will only consider those situations in which the speed changes in a regular way. Otherwise simple arithmetic is no longer sufficient to describe the motion, and we want to keep it simple.
4.3.1. acceleration is change of velocity divided by time
We will define acceleration as the time rate of change of instantaneous velocity.
Suppose that a car accelerates from rest to 60 miles per hour at a constant rate. Then we might measure its instantaneous velocity each second and record it in a table. We could measure the instantaneous velocity by taking a photograph of the speedometer each second, although this is not the only method.
Suppose that it takes the car exactly 12 seconds to reach a speed of 60 mph, starting from rest. The table of speeds would look like this:
What do these numbers tell us about the speed and the rate at which it changes? Let's see.
The car went from rest (0 mph) to sixty miles per hour (60 mph) in twelve seconds (12 sec). So its rate of change of speed was sixty miles per hour in twelve seconds or five miles per hour per second. (60 mph/12 sec = 5 mph/sec). This means that at the end of each second it was traveling five miles per hour faster than at the end of each previous second.
Sure enough, it we look at the table, that is exactly what happened. The velocity increases by 5 miles per hour for each second. We might say that five miles per hour is added to the velocity each second.
We call this uniform acceleration because the rate of increase of speed is constant.
Notice that we have used the concept of instantaneous velocity at each second although the units are in miles per hour. We might also say that the car accelerated at the rate of five miles per hour per second, or equivalently 7.33 feet per second per second (88 ft/sec divided by 12 sec).
Notice how the unit of time is compounded in this representation. It is used twice, once to indicate the velocity (rate of change of location) and again to indicate the acceleration (rate of change of velocity).
This compounding of the time unit or any other unit has no precedent in any recorded thought before Galileo. It is a sophisticated concept born of genius, one of many such strokes on the part of Galileo.
It is preferable, to keep things simple to speak of time in the same terms. We prefer to think of it as 7.33 feet per second per second rather than 5 miles per hour per second.
Note how the unit is constructed. Feet per second per second really means (feet per second) per second. We normally abbreviate this as feet per second squared, so we would say the acceleration of the car was 7.33 feet per second squared.
4.3.2. acceleration is the time rate of change of speed
Now lets turn our attention to a graphic way of visualizing these relationships. What we are about to see is one of the more amazing mathematical phenomena. It is indispensable to our modern science. That such relationships exist between numbers and pictures is amazing enough. That we can use the same relationships to describe and visualize something abstract like motion is even more so.
The type of graph we will use dates back to the early part of the seventeenth century and Rene Descartes. Descartes was a French expatriate, self-exiled to Holland to avoid the irrational wrath of the Inquisition in France. In Holland the climate was much looser. There the concern was more in making money and establishing trade than in persecuting and punishing people for believing in the wrong things or for having the improper ideas.
Descartes formalized the process of analytic geometry which allows for a relationship between equations and graphs on a Cartesian coordinate system.
You have probably seen these Cartesian graphs before. Don't worry, we will be using them in a much different way than you might have in Algebra II.
Our interest in the graphs is the relationship between the quantities which are plotted, not in the plot itself.
4.4.1. Cartesian Coordinates
The Cartesian coordinate system is two number lines which intersect at a point called the origin. Each line is called an axis of the graph. The scale on the two axes may be the same, but it doesn't have to be.
Numbers can be measured off on the two axes and a point plotted which represents the distance from the origin in both directions. For example we might choose a graph on which to plot the measured pairs of values of time and distance from the above table (section 4.3.1). One of the data pairs is plotted on the graph below. Can you determine which one it is? If you can not, then take a few minutes to figure out how the point is plotted before continuing.
With this understanding of the graph, its coordinates and the way a point is plotted, we are ready to look at relationship between the numbers and the shape of the graph. Before we actually study this graph we want to back up a little and consider shapes of graphs and the relationships they reveal.
4.4.2. Constant velocity
First let's look at a graph of distance and time for an object which is moving at a constant speed. On this graph we are using Cartesian coordinates but we have not put numbers on the axis. We do this because we want to illustrate the nature of a general relationship between distance and time. The coordinate axes are simply shown with arrows which indicate increasing distance and time.
By definition, a constant speed means that the same distance is traveled in equal time intervals. If the speed is constant, then the relationship, or ratio between distance and time remains the same, as shown on the graph.
You will notice that this constant relationship between distance and time creates a straight line graph.
The size of the interval of distance compared to the interval of time is a measure of the speed.
Consider an object moving at a faster constant speed. It will cover a greater distance in a given time interval than the slower object. How would the graph of the faster object compare with that of the slower object? What would the graph of a stationary object look like? What would a vertical line on the graph indicate?
4.4.3. Constant of proportion
The number which represents the ratio between the two quantities plotted on the graph can be characterized in several ways. First and foremost it is the number which must be added to the vertical axis for each interval on the horizontal axis. In our example above, an acceleration of 5 mph/sec means that 5 mph is added to the velocity each second. The number and corresponding unit, 5 mph/sec, is the constant of proportion. It is also the number which when multiplied by the horizontal quantity gives the corresponding vertical quantity. We saw in an earlier lesson that multiplication is nothing more than repeated additions. Here is a concrete example of that relationship.
Note that there is both a qualitative and a quantitative relationship here. Qualitatively, we can say that when time increases, velocity will also increase in direct proportion. The term direct proportion means that the constant of proportion, whatever the number, is unchanging. Another way to say this is that the two quantities have a constant ratio and plot as a straight line on a graph which trends upwards. Another way to visualize the direct proportion is to note that if one quantity increases by a certain factor, the other increases by the same factor. For instance, if the time increases from 4 seconds to 8 seconds ( a factor of two) the velocity increases from 20 mph to 40 mph (a factor of two). If you examine the table you will see similar relationships for a factor of 4 (from 2 seconds to 8 seconds), 5 (from 2 seconds to 10 seconds), and so forth.
The constant of proportion is also represented quantitatively by a number which represents the ratio, in this case the number five.
We could say for this graph that it is qualitatively a direct proportion with a quantitative constant of proportion equal to 5 mph/sec. The constant of proportion on a linear graph is quantified by the slope (rise over run) of the straight line on the graph. Changing the rate of acceleration will change the numbers but will not alter the type of relationship between the numbers as long as the acceleration is uniform.
The role of a constant of proportion can be illustrated with a simple example.
Suppose an item at the store is sold by the pound (or by any weight unit), such as fresh fish. The act of selling by the pound (with no discount for large purchase) is a direct (linear) proportion. Obviously, the more you buy the more it will cost, and if you buy twice the weight it will cost twice the money. This is the obvious qualitative relationship.
To know exactly how much fish you can buy with a certain amount of money, you have to know the price per pound. This is a ratio of cost to weight which is a constant of proportion. So if the fish is $6.00 per pound then you can easily figure the cost for any amount of fish. It doesn't matter whether you buy one pound, two pounds, or a fraction or a decimal of a pound. The cost will always equal the price per pound times the weight in pounds. If the price changes, it does not alter the qualitative nature of the relationship. You can still buy twice as much fish with twice as much money. All that changes is the actual amount of fish you can buy with a certain amount of money.
This concept of a constant of proportion is extremely important in understanding the principles that we will be considering in future lessons. To be sure that you understand it, think of other examples of direct proportion, such as the number of revolutions of a car's tire compared to the distance traveled. Can you think of others?
4.4.4. Irregular velocity
Here is a graph of an object moving at an irregular velocity. The line is not straight. You can see that the steeper the graphed line the greater distance is covered in equal time, or the faster the speed.
From this information we can state a general principle: On a graph of distance and time, the steeper the slope of a line, the greater the velocity it represents. If the line is straight, it indicates a constant speed, if it is curved it indicates a changing speed.
4.4.5. Regularly Increasing Velocity
In this figure we see the graph of an object which has a regularly increasing velocity. Notice that in each successive time interval the distance traveled increases a little more than in the previous interval. This produces a smooth upward curve. Without further proof we will state: A graph of distance and time which represents uniform acceleration will curve upwards in the shape of a parabola. The parabola is a conic section. A graph in the shape of a parabola shows a second power relationship. When the parabola bends upwards, the relationship is between the first power of the quantity plotted on the vertical axis and the second power of the quantity plotted on the horizontal axis. In this case, for uniform or constant acceleration, distance is proportional to the second power of time. We say, distance is proportional to time squared.
The curve is not exactly a parabola in this graph because we have plotted average velocity over a fairly large time interval rather than instantaneous velocity.
Now we are ready to look at the graph of velocity vs. time for uniform acceleration.
4.4.6. Velocity vs. time graph
Now let's look at the graph of our velocity and time data for the accelerating car. What shape would we expect the graph to have? Let's guess first, then see if we are correct.
In our graph of constant velocity above, we saw that when the relationship between distance and time was constant, the graph plotted as a straight line. Is this true in general? If so then the graph of velocity vs. time for our accelerating car should be a straight line because we know that the velocity changes at a constant rate. So we expect a straight line on our plot of velocity and time for uniform acceleration.
You might want to plot the data pairs in the table on the graph above before you look at our graph.
You will see that the points fall on a straight line, just as we expected.
This should not really be surprising, for the same reason that it should not be surprising that a flat board laid on a staircase should touch each of the stair treads, assuming the board is straight and the stairs are constructed properly with each tread and riser exactly the same size.
If the same increment of velocity is added in each equal time interval, then the relationship between the two, like the relationship between the tread and rise on the stairs, is a constant one.
188.8.131.52. Instantaneous velocity vs. average velocity
Now consider the following case. A car sits at rest at a traffic light. Another car approaches the light in the adjacent lane, moving at a constant speed. At exactly the instant that the light changes two things happen. The oncoming car, call it A, is beside the stationary car, call it B, just as the latter begins to accelerate.
Now suppose that car A continues at a constant velocity, and car B accelerates at a constant rate such that they both reach the next intersection at exactly the same instant.
If you have difficulty visualizing this situation,refer to the video program for visual reinforcement, or look at a streaming movie of the animation.
There are several questions we might ask about the motion of the two cars and the graphs of their motion in the way of analysis of this situation.
1. What is the average speed of the two cars.
2. How does the final speed of the two cars compare.
184.108.40.206.1. Average Speeds
From our definition of average speed, total distance divided by time, it is clear that the two cars must have the same average speed. They travel the same distance in the same amount of time. Their average speeds (velocities) must be the same according to the definition of average speed. How can this be? Obviously the details of their motion are very different. Car A moves at a constant speed while car B accelerates from rest to some final speed, achieved at the next intersection. A graph of the velocity vs. time for the two cars can help to illuminate the situation. The graphs are shown in the following figures.
The rectangle is formed by the constant velocity of car A. The horizontal line labeled "average velocity" represents both the instantaneous and average velocity since the speed does not change.
Here's the graph of motion for car B.
The large triangle represents the motion of car B. Initially it is a rest (it has zero velocity). At time zero (when the light changes) it begins to accelerate, reaching the intersection at the exact same time as car A.
220.127.116.11.2. Final Speeds
For the first half of the elapsed time interval, car B is traveling slower than car A, but at exactly halfway in the time interval (halfway in time, but not in distance. Why?) their instantaneous velocities are the same.
What is happening during the first half of the time interval? What would you see if you were in car A? What would you see if you were in car B?
In order to reach the next intersection at the same time, car B must spend the second half of the time interval catching up to car A. By the time they reach the intersection, at exactly the same instant, the instantaneous velocity of car B must be exactly twice that of car A.
Why twice? Let's compare the graphs of the two cars.
Look at the two small triangles (blue and pink) in this figure. They are congruent, which means that their two vertical sides are equal in length. That means that the average velocity is exactly halfway between the initial and final velocities of car B. That means that the final velocity of car B must be exactly twice that of car A.
On the graph, the rate of acceleration can be represented by the slope of the line. By definition, the slope of a graph is the rise divided by the run. In this graph the rise represents the change in velocity and the run represents time. If we call the slope a, then symbolically, a = [[Delta]]v/t (the [[Delta]] symbol means "change of").
If you are still with us, you will recognize that this is exactly the definition of acceleration. You will also note that the horizontal line, representing the constant motion of car A has a slope of zero, indicating that the acceleration of car B is zero, or that it is not accelerating, or that it is moving at constant speed. All three statements are equivalent.
Can you see from this graph that the area of the triangle and the area of the rectangle are the same. If you take the smaller triangle on the upper right and flip it (do it mentally, or make a cutout) you will see that it is exactly congruent with the other small triangle in the lower left. Congruent means they are the same size and shape. This is a good word to look up in the dictionary if you are not familiar with it. We will come back to the meaning of the area in awhile.
If the slope of this velocity vs. time graph represents the acceleration, what does the area represent? Let's see.
To calculate the area of the rectangle on the graph you would multiply the length times the width. On the graph the length is time and the width is average velocity. So multiplying l x w is equivalent to multiplying v x t.
But wait. Velocity multiplied by time equals distance. Sure it does. Look at the definition of average velocity: distance divided by time. Symbolically v = d/t so d = vt.
The area of the rectangle equals the distance travel by car A in the given time interval. Now, we saw above that the area of the rectangle and the area of the large triangle (representing the motion of car B) are equal.
Do you see the point. Here's the logic and the only conclusion we can reach.
The area of the rectangle represents distance traveled by car A.
The areas of the two graphical figures are equal, and the cars travel the same distance, therefore the area of the large triangle must represent the distance traveled by car B.
18.104.22.168.5. Putting it together
What we have just discovered, the relationship of graphical geometry to numbers and equations is the basis for all of our modern physics. There are relationships between the variables we call time, distance, velocity, and acceleration. These relationships are mathematical and graphical As we will see, Newton used these relationships when he invented calculus. These relationships between slope and area are general relationships and form the basis for differential calculus (slopes) and integral calculus (areas), which became the backbone of physical science after Newton showed us how they work.
22.214.171.124.6. Slopes, Areas, and Physical Reality
In general, the slope of a graph is a ratio of the two quantities being plotted. The area of the graph is a product of the same two quantities.
On a graph of velocity vs. time, the slope represents acceleration and the area represents the distance traveled.
The fact that the slope and areas are related in this way for certain physical quantities has got to leave us wondering why it should be so.
The only answer is no answer at all: Whether we like it or not, physical reality is based on relationships and described by mathematics. We can use graphs and equations to represent change, contrary to Aristotle's teachings. This is one of the greatest revelations of modern science.
Pythagorean? Somewhat, but more than that. We distinguish between coincidental relationships and meaningful ones, although admittedly we can't always tell the difference at first glance.
126.96.36.199. Algebraic relationships between distance, velocity, acceleration, and time
4.5.1. average velocity is total distance divided by time
4.5.2. instantaneous velocity is average velocity over a suitably small time interval
4.5.3. acceleration is change of velocity divided by time
4.5.4. on a graph of velocity vs. time the following are true:
188.8.131.52. constant or uniform acceleration will plot as a straight line
184.108.40.206. the slope of the graph represents the velocity
220.127.116.11. the area of the figure formed by the acceleration graph represents the distance traveled
Our modern view of motion is very different from that of the Scholastics, whose ideas were borrowed almost directly from Aristotle. Aristotle had claimed that mathematics was of no use in describing change. In his view all change, including motion, was connected to the cosmology of imperfection.
Aristotle described four kinds of motion: alteration, local, violent, and celestial. Each of these had different properties and different causes.
One of Aristotle's greatest errors was in his views on motion. He believed that all motion must have a cause, and that motion could not continue without some impelling force. He attributed the cause of motion to the Prime Mover, or to the desire of the four elements to seek perfection in their logical place in the sublunar realm. Part of the arguments against a moving Earth were based on Aristotle's incorrect analysis of motion.
Galileo recognized that if he could prove Aristotle's views on motion to be incorrect then it would be easier to convince people that the Earth could move. If Aristotle could be wrong on one account then he could be wrong on others too.
Our modern view of motion derives from Galileo's work, which we will study extensively in future lessons. By precisely defining the parameters, or variables of motion, Galileo was able to show logically and mathematically the relationship between those variables. He did this in such a way as to be able to make measurements of motion in terms of distance and time and show conclusively that acceleration due to gravity was uniform.
Our modern view incorporates the use of the Cartesian coordinates to show the relationship between time, distance velocity, and acceleration. From these tools we can see that constant acceleration produces a straight line on a graph of velocity vs. time. Furthermore we note that uniform (constant) acceleration produces a direct relationship between distance and the second power (square) of time. We can also show that the slope of this graph represents acceleration and the area of the graph represents distance.