## Extended Syllabus

The following describes and discusses the labs listed in the syllabus of "Foundations in Science." It draws from Dr. Shahn's paper Foundations of science: A lab course for science majors, published in D.E. Herget (Ed.), More history and philosophy of science in science teaching, Proceedings of the first international conference (pp. 311-352). Tallahassee, FL: Florida State University.

Theme I: The Heliocentric Theory and the Study of Motion

Lenses and Telescopes:

The first lab in the course is a prototype of the learning cycle approach. Students are given a set of eight lenses to "study". They are a mix of concave and convex lenses of varying focal lengths. The discussion at the end of the exploration phase shows, aside from some "accidental" or "irrelevant" observations (one of the lenses is greener than the others, they have different diameters), some lenses form images and others don't, and that the sizes of the images differ. The ability to form images is associated with the general shape of the lens (convex or concave).

It is also noted that various combinations of the lenses have different effects. Some (concave) lenses which don't form images exactly counter the effect of some that do, so that looking through such a pair has the effect of looking through a plain piece of glass, and pairs of convex lenses seem to make each other "stronger". These are to be exploited in parts II and III of the learning cycle.

The discussion is also used to get the students to "invent" a simple optical bench that can hold a lens in place so that the distance from the image it forms of a distant object can be accurately measured. When this is done, the distance (focal length) is then used as a means of characterizing the lens, (positive notation) and the lenses which exactly balance them are called "negative".

The final application phase of the learning cycle has the student experiment more systematically with combinations of lenses and distances between them. This is first done holding them by hand, and then on the optical bench where it is easier to make sure that the lenses are aligned and kept a fixed distance from each other. By the end of the lab all students have invented a telescope.

This lab is placed at the beginning of the course, and not where it might belong historically (with Galileo's use of the telescope), because it provides an opportunity for students to interact with objects and phenomena in a context-free manner. There are also a number of administrative reasons for using this lab. The most significant of these is that it serves as an introduction to the laboratory as a place and to what we do there. In this sense it takes the place of a roll call and equipment check-out which is the experience of many initial labs.

Celestial Phenomena of the Night Sky:

The second lab provides an occasion to make sure that all students know what may be seen in the sky at night. These most primitive observations are the experiential basis of the first story line. It is well known that "light pollution" makes it difficult to see stars in an urban environment. But beyond that, the view of the sky is so obscured by buildings that it is also difficult to see the moon, and many students are unaware of the phenomena of its phases, and how these are correlated with when it rises, or the changes in its times of rising. Also, while students may be very well acquainted with the idea of the signs of the zodiac, the relationship of these to constellations and the apparent motions of the sun against the fixed stars, or the wanderings of the planets among them are foreign concepts.

Although the goals of this lab might be met by using a dedicated planetarium, the technology of even a simple instrument might be intimidating or overshadow the points to be made. Our approach is to use flashlights in a darkened room to indicate the relative motions of the various stellar objects as we consider them. In this lab we are not concerned with explanatory models of what can be seen. Rather, we are particularly concerned with having students realize that all (but one) objects seem to move around the earth roughly once each day, circling the pole star. The sun and the moon, however, seem to slip in this motion, and this regularity requires a degree of modification. Finally, the planets show some sort of modification even in this slipping motion, and at times they seem to go forwards as well as backwards in what has been called a "retrograde motion". We also use this lab to point out that apparent position may best be described in terms of angular measure.

Archimedes' Principle:

In terms of the lectures this lab, too, is a bit out of place, but our decision was to keep the next lab in closer contact with the flow of ideas. By studying Archimedes' Principle we show that at least one Greek was a master of experiment. (That he was also a master of analysis well ahead of his time is fascinating, but not relevant.) In addition, in our presentation there are several fundamental ideas that we wish to cover in this lab. These include the primary concepts of weight and volume, as well as the derived concepts of specific gravity, density, and buoyancy. While volume may well be "primary", the fact that it cannot be easily measured in all cases is one of the major features of Archimedes' achievement. In addition this is the time when students are introduced to the operation of a simple balance, the fact of experimental error, and the idea that repeated measurement and averaging are one means of reducing this.

In fact we use three different ways of determining specific gravity based on Archimedes' Principle. The first uses a metal cylinder which exactly fits into a container. The volumes of the cylinder and the inside of the container appear to be equal by inspection. The weights of the cylinder and an equivalent volume of water must then be determined by weighing "container + cylinder" and "container + water" and subtracting the independently determined weight of the container. The reasoning appears to be straightforward, but for many students this is a struggle.

The second method follows the idea that a submerged object displaces its volume of water, and uses a spill can to collect the water which overflows from a brim-full container when the same cylinder is immersed. This water can then be weighed. The third method involves determining "loss of weight" of the cylinder when it is immersed. In principle the weights of the water should be the same in the first two methods, and equal to the observed loss of weight of the cylinder in the third method. In fact they are not, and careful observations accompanied by discussion will be needed to find the possible sources of error.

The use of three different approaches to the same problem not only introduces the idea that there need not always be one "right" way to do something, it also provides the excuse to use the balance several times, and thus to acquire familiarity with the act of weighing. While young children may be quite content to repeat processes until they master them, college students are often impatient if they cannot quickly perform a given task.

As well as the idea of averaging ("average" defined as to make better estimates of such values as individual weights, the notion of precision is introduced in this lab. All weights can be determined to three decimal places, but the procedure of division can be carried out indefinitely. Why we stop after three places in this process is the subject of more discussion. We also introduce the concept of "average deviation", which we define as a measure of error." Some of these concepts are used in subsequent labs, but even if we do not use them, we feel it is important that students get the idea that there are ways of obtaining measures of error as well as the actual values of weights.

Finally, with the idea that each kind of material has its characteristic specific gravity, we are able to use this as a means of determining the composition of an "unknown". This, of course, was the reputed motivation for Archimedes' discovery. This use, however, is not trivial. In the first place the idea that the volume or weight of an amount of gold should not change as it is fashioned into a crown is one of the first recorded applications of the Piagetian idea of conservation. Since this is considered to be a landmark in science, there is no reason to expect our novice science students to grasp it immediately. In the second, the concepts of specific gravity (or density) are derived quantities that cannot be determined as the results of single measurements as can weight or length. Nor can they be as directly computed as volume can, for simple shapes. Rather, as intensive parameters they represent properties of the material, not the particular sample that is being studied. This is a point that many students do not readily appreciate.

Measurements in Space:

A subtitle of this lab might well be "Geometry and Indirect Measurement". In the context of the course, one of the first "turning points" or major advances which took place in the ancient world was the ability to put some sort of scale on the universe. This lab is the occasion to make clear what the real significance of this achievement was. In general it is not clear whether in cases such as this the experience should precede the lecture, or vice versa. In one case it can be argued that the experience makes the lecture discussion more clear, in the other the argument is that the setting provided in lecture will provide better motivation for the lab. We have seen justification for both approaches, and thus rigorously advocate neither.

The aspect of geometry that is crucial to any sort of indirect measuring procedure, whether the relative distances of the sun and the moon, the size of the earth, or the height of a pyramid, is the concept of similar triangles. For students who never mastered this in high school, or who may never have had the standard course in geometry, it is not enough simply to say what the properties of similar triangles are. We show them by using a set of fifteen colored paper triangles, some pairs of which are similar. The use of colored paper makes it possible to specify identical triangles without knowing which properties are critical. Then, as they are compared by superposing one on an other it becomes obvious that similar triangles have equal corresponding angles, even if it is necessary to flip one of them over so that the correspondence is self evident. Several pairs of right triangles are included in the set so that it is seen that not all triangles with a right angle are similar.

Now, by measuring corresponding sides it is possible to demonstrate that the ratios are equal. This is done in both ways; the ratio of the large side to the small side of one triangle is the same for any similar triangle, and the ratio of the large side of one triangle to the large side of a similar triangle equals the ratio of the two corresponding small sides. It must also be pointed out that the "constants of proportionality" in these two cases are different. (If A, B, and C are the sides of a large triangle, and a, b, and c are the corresponding sides of a similar smaller triangle, the equalities of ratio would be: a/A = b/B = c/C, and A/B = a/b.) While these operations may seem trivial, they are formally at the heart of the proportional reasoning process which is one of the major stumbling blocks for many people.

This manipulation, with the accompanying discussion, takes about a half hour, and prepares the way for consideration of different ways of determining distances indirectly. The first of these is the classic "rule of thumb" which involves noting the apparent shift in position of a thumb held erect at arm's length as it is viewed against a distant structure alternately with the right and left eyes. The similar triangles used here are determined by the inter-eye distance and the length of the arm in one case, and the apparent shift in position and the unknown distance in the other. When there is some way of determining the actual distance indicated by the shift (as thirteen window widths, with the width of a window being about three feet), it is then possible to estimate the distance to the building.

Subsequently, distances are determined indirectly to the far corners of the lab by noting the angles made to these points from two different sighting points that define a base line which can be measured. Similar triangles are made on a piece of paper, and the unknown distance is determined by measuring on this diagram. It is confirmed by measuring in the lab. Finally, a method that uses the concept of parallax is used. In this, the apparent angular shift in position of one distant point, say the window frame, against the background of a building or distant trees, is determined, and using the above method to determine the distance to the nearer point, one can then estimate the distance to the further point.

With this much work it is felt that the students have developed an idea of what it means to measure indirectly, and have some confidence in this. The remainder of the lab is spent discussing several geometric proofs that have been presented in lecture and reading. These involve Aristarchus' argument dealing with the relative distances of the moon and the sun from the earth (based on the time differences between first and third quarters, and third and first quarters), his arguments of the relative sizes of the sun and the moon, Eratosthenes' determination of the size of the earth (based on the differences in the angular height of the sun at noon), and the significance of Tycho's inability to detect any parallax in his observation of a nova. While all the students should by this time have heard and read these arguments, recreating them in small groups in the context of their own experiences with indirect measurement serves to remove the conclusions from the abstract "to be memorized" category to the more real world of their own creation.

This is one of several instances where we attempt to have the students reproduce several stages of a complex argument with continual supervision. Not surprisingly, many of these reasoning strings involve proportional reasoning. As has been noted above, many students have difficulty grasping the rudiments of simple proportions, but these are often straightforward to demonstrate. What is more difficult is mastering the idea that proportions may have to be used sequentially. Again, the manipulation of the symbols or the numbers is easier to perform than establishing and justifying the sequence.

Models and Their Consequences:

It is virtually impossible to discuss the apparent motions of the stars and planets without invoking some sort of model. In its simplest form the model will call to mind an object that bares a degree of similarity to the features of the sky that are under consideration. Thus the ancients' reference to the sky as a plate or dome conveys the idea of a covering, but also carries the notion that all of the stars are on one surface. This is consistent with what one sees with little additional apparatus and no significant measurement. The extension to a "celestial sphere", which is still in our contemporary working vocabulary, does not appreciably change this. It requires a fair bit of reasoning to appreciate that two objects that are both so far away that neither would exhibit parallax would appear to be equally distant, even if one were twice as distant as the other. This is the kind of realization that this lab is intended to convey.

On a more mundane level we concern ourselves with the Ptolemaic and Copernican models of the planetary system. It was, of course, the battle between these two models which is considered to accompany the birth of modern science, and which, following Kuhn, has been used as the paradigm of the use of "paradigm" in current discussions of revolutions in science.

We use commercially available models of the celestial sphere centered on the earth, and of Ptolemaic and Copernican planetary systems. In addition we use a large Styrofoam sphere attached to a stick so that the explanation of the phases of the moon as due to its position relative to the sun can be visualized with the aid of a localized light source. The use of all of these models in the context of discussion develops the major point that one may often account for a limited set of observations as resulting from different -- and possibly conflicting -- physical models.

Once this point is made clear, and the fact is accepted that a model may only be considered if it does account for many observations, the question is raised, by what criteria does one choose between contending models? A key point here, as historically, is how one accounts for retrograde movement of the planets. The Ptolemaic model is acknowledged to have descriptive/predictive value, but the reality of the various kinds of motion that are required is clearly ad hoc. Moreover, if one adds to the consideration the fact that retrograde motion occurs only when the planet in question and the sun are in opposition, there is a feature that follows from -- or can be accounted for by -- the Copernican model but has no place in the Ptolemaic system.

The general conclusion, of course, is that when properly considered, models do have consequences, and ultimately it is these that enable one to choose between them. Despite the apparently obvious nature of this conclusion, many students still do not appreciate the different roles of models and observations on our road to understanding. Just as they will frequently give examples when asked for definitions, they confuse the model with the object being observed, and limit their thinking to the account of specific observations. We have seen this in papers that students have been asked to write, and note it here as a point that instructors should be aware of as they use models for any sort of explanatory purpose.

Galileo I: Uniform Acceleration c. 1600:

This lab is the first of a set of four which are closely related, the results of one being built on in those that follow. In our implementation of the course the substance of these four labs, jointly with the related lecture and reading material, also form the focus of one of the major papers that we require. For this reason, while it is still essentially self contained, this lab should be viewed in terms of what follows.

Galileo is generally credited with introducing the experimental approach to modern science. Specifically his experimental studies of motion are considered revolutionary. Consider, for example, the idea of uniform motion. In a straight line, it is almost trivial to say that this refers to equal distances being traveled in equal time intervals. Equivalently, one could say that equal times are used to traverse equal distances. The primacy of time as an independent variable is not evident.

When it comes to the idea of changing motion, however, the meaning of uniformly accelerated motion is not so obvious. For instance, should one say that a falling ball, which obviously does accelerate, acquires equal increments in velocity for each unit length? or for each unit of time? Before Galileo it had already been argued that the first of these explanations could not be true because then an object at rest, never moving through the first unit of length, would never accelerate, and thus would never move. But does this mean that the second explanation must be true? An Aristotelian, having ruled out one of two possibilities, would accept the other. For a logician, this may be appropriate. For Galileo as a student of nature, it was not.

Galileo reasoned that the remaining conclusion would have certain consequences, and he set out to test them. One of these is expressed algebraically in the form s is proportional to t2; that is, in twice the time the ball will roll four times the distance. We do not expect our students to recreate Galileo's reasoning, but we do ask them to follow it so that they see why the experiment they will perform is designed the way it is. In part Galileo had to slow motion down; tracking an object in free fall with regard to where it is and when is difficult even today. Galileo decided to use a ball rolling down an inclined plane, where it is clear that the less the incline, the more slowly the ball changes its speed.

(In a sense, Aristotle, who is known to have spent much time near the shore studying marine life, may have approached the idea of "slowing motion" by watching weights fall through water. Interestingly, different weights do fall through water differently, with heavier ones falling faster! This may be easily confirmed with a graduate filled with glycerine, or with stop frame video in as little as a large beaker of water. In fact we use this demonstration to justify the need to continue with Galileo's work.)

Having reached this point, the question has now to be asked, how do we obtain the time and distance measurements needed to analyze the nature of motion down the plane? Obtaining a flat plane or straight groove is no simple matter. Galileo noted that he worked hard at his task, and even having resort to the results of such modern technology as extruded aluminum bars does not immediately give optically straight paths. We finally obtained straight tubes which we bolted together and then fastened to two by four beams. The bars are shimmied to make sure that they are straight.

As for the time, we note that Galileo had at his disposal no time measuring apparatus beyond that available to the Greeks, the klepshydra, or water clock (literally, water thief). Indeed, insofar as time measuring technology is concerned, Galileo is credited with having noted the isochronism of the pendulum, even though he did not invent the first clock. In introducing this apparatus, which we fashion out of a carboy with a narrow orifice, we are forced to point out the assumptions about the nature of time which are made, and the fact that by weighing the amount of water that is collected between "start" and "stop" we are measuring a time interval indirectly, by virtue of a phenomenon that we believe to proceed in a certain way. Indeed, we are (re)defining time. Of course, this is essentially the case regardless of what clock or chronometer is used, but the point is much harder to appreciate with a digital wrist stop watch whose mechanism is totally obscure, yet which is readily accepted as providing a measure of time.

Using the water clock poses other problems. Galileo reported that his time measurements were correct to a "tenth of a pulse beat." What does this mean? The balance, which has already been mastered in the Archimedes lab, and which provides the measure of time with the klepshydra, is accurate to several places. How many of these are significant? As the same event (a ball rolling a fixed distance from rest) is timed repeatedly, it becomes apparent that there seems to be an experimental error which is greater than the precision of the weight determination. This is easily understood to be due to the reaction time needed to start and stop the collection of water as the ball passes two fixed points. This is known to be about a tenth of a second, and hence in accord with Galileo's result. At this point it may even be appreciated that the use of a digital watch with a 1/100 second accuracy would provide no better results for this experiment, as it would still have to be started and stopped on cue.

Thus in only the introductory part of a lab we have (1) defined time as being continuous and "flowing", and hence being susceptible to measurements in fractions of the discrete units (pulse beats) used in every day discourse; (2) we have introduced an indirect means of making these measurements; and (3) we have made the distinction between precision and accuracy. The extent to which any of these is pursued in discussion is of course optional.

At this point the experiment itself proceeds in a straightforward manner. The length of the groove is marked with 16 equal spaces. The times needed to roll 1, 2, 3, 4, ..., 10 of these spaces are determined, and the question is asked, "How far does the ball roll in twice (or three times) the time that it takes to roll one distance?" The answer is clear, but not without experimental error. Some groups in each class perform better than do others. With the "law" discovered, we can then ask for a prediction: "How much time does one expect to elapse for the ball to roll 16 spaces?"

The last point to be discussed in this lab is a speculative question, "Why didn't the Greeks discover Galileo's law of uniform acceleration?" The techniques he used were available to them, so it wasn't a matter of technological limitation. The answer is clearly philosophical. Simply put, the Greeks believed in ideals (not the accidents of special cases) and that understanding was a process that was intellectual (and hence could not be significantly enhanced by appeal to experience or experiment). Related to this is the oft-stated notion that the Greeks considered experiment, in the sense that it was hand work, the task of slaves, not thinking people. (Archimedes lived in Syracuse and hence was apart from the major achievements of Greek thinking in space as well as time.) This lab thus becomes the occasion to note that science may be limited by a point of view or frame of mind as much as it may be by lack of information or incorrect concepts. Not only is the point made, but it is appreciated by a number of students. A reasoned discussion of this has appeared in a number of the student papers which deal with this lab.

Galileo II: Uniform Acceleration c. 1980:

This exercise repeats many features of the previous one with the use of a more modern apparatus. We use an air track in which a glider is suspended on a cushion of air to eliminate friction. Time intervals and positions are simultaneously indicated on a strip of paper which is marked by a spark that goes from a wire attached to the glider. The flatness of the track is first demonstrated when it is horizontal, and it is seen that from rest the glider goes to neither side. The uniform nature of the timer is demonstrated by observing that in this horizontal position the marks made by the spark are equally spaced. The approximate precision of the time interval is 1/10 of a second, and distances of about 1/10 of that between dots can fairly easily be determined. Inspection of the apparatus in operation shows that the spark seems to jump around in the air, and not always take the same path. This accounts for some error, but not appreciable.

This repetition of essentially the same experiment is performed for a number of reasons. In the first place, it reinforces the conclusions reached the previous week, and permits questions of understanding the mechanical aspects to be addressed in a slightly different context. It also enables us to go over the mathematical reasoning again. The use of an obviously more sophisticated device has some attractions, as well as being possibly intimidating. Since the nature of the experiment should be clear, the students can look at the apparatus to see how it does its job. We will be using the air track in the next two labs, so that as they are approached the apparatus should not be the sticking point, and the concepts to be discussed there will be more easily accessible.

Clearly, having moved forward almost four centuries, it is no longer reasonable to ask that a long-known law be rediscovered. Instead, we speak in terms of verification or confirmation. Interestingly, because the sparking does not start when the glider is released, we cannot ask how much time elapses between rest and specified positions. Rather, we return to the idea of uniform acceleration discussed above, and raise the question, "How much distance is covered in equal times?" In this case equal times will be marked by equal numbers of marks, and the distance covered is obtained simply by measurement. The apparatus works quickly, and each student is able to have his own strip of paper for analysis. The definition of uniform acceleration we are using is simply one involving equal increments of velocity in equal times, and this would be represented by equal increments of distance. Thus testing the prediction is fairly simple.

Just as an air track eliminates friction for motion in a line, so an air table can be used to eliminate friction for a puck moving in two dimensions on a plane. This makes it possible to approach the problem of considering the motion of a projectile which was also a classic problem in mechanics and was discussed by Galileo. When tilted, the table provides a way of slowing down the acceleration due to gravity. The difficulties inherent in obtaining accurate simultaneous measures of position and time are easily appreciated by playing with this device.

Our contemporary means of analysis has been to make a video tape of the puck falling from rest, and moving with different initial velocities at different table elevations. We then view this tape with a player capable of single frame advance which provides excellent still images. The table surface is marked with a grid. This enables the class to determine horizontal and vertical positions at the same instant, as they keep track of the number of frames between measurements. For a given plane elevation we are interested in determining the rate of fall from rest, and then both the vertical and horizontal components of motion for different initial velocities. This exercise demonstrates that the vertical fall does not vary with the initial horizontal motion, and thus that the vertical and horizontal motions are independent.

For each time we also have the students plot the horizontal and vertical positions of the puck on a set of axes which represents the grid on the surface of the air table. The result is a detailed picture of the trajectory. We also have them plot the horizontal and vertical positions as a function of frame number, or time. For many students this is the first exercise in graphing that they can recall. The nature of the apparatus and the data are such that the meanings of the graphs are clear, and for this reason it is easy to tell by inspection whether they are correct. The use of graphs will be extended in subsequent labs.

Newton: The Second Law F = ma:

By the time our students confront this lab they have already been given a discussion of the meaning and origin of Newton's laws. Thus we use the air track as a means of testing the second law. The standard apparatus provides gliders with masses in the ratio of 1:2:3. It also has a pulley at one end which makes it possible to attach a small weight to the glider with a thread.

Our experiment is divided into three parts. In the first we use different forces with the lightest of the three gliders. The forces are provided by using different numbers of paper clips; we are not concerned with absolute numbers, only with the proportional effects. We end with the question of whether there is any relationship between the acceleration (measured as in the previous lab) and the number of clips.

The second part examines the relationship between a given accelerating force (fixed number of clips) and glider mass, which is varied by using the set of gliders available. The third part of the lab tests the combined generalization, which is Newton's second law, for combinations of force and mass which have not been used previously. The students create a table in which measured (or predicted) accelerations are entered in columns headed by relative masses (1, 2, and 3) and in rows identified by relative forces (numbers of clips).

The experimental parts of this exercise are deceptively simple. The understanding rests on fairly thorough grasp of proportions, and this is something with which many of our students need considerable assistance. Thus, in Part I the observation is that as F increases, so does a; the conclusion, then, is that a is proportional to F. In Part II the observation is that as m increases, a decreases; numerically this is seen to be an inverse proportion. We have noticed that going through this exercise in the lab is not sufficient to insure that students retain the concept of the second law, and while they may be able to substitute numbers in the equation, they cannot apply the law even in simple situations analogous to the lab.

Conservation of Mechanical Energy:

The various concepts involved in the understanding of motion are by no means self evident. Force, work, energy, momentum, and acceleration have all at different times assumed center stage. And all have their common usages which are not always in accord with the technical definitions. Today energy is in many way of paramount importance, but the conservation one reads of in the papers is not the same as that which, in the first law of thermodynamics, was one of the high spots of 19th century science. This lab is concerned with the idea of mechanical energy, the fact that it can exist in two distinct forms (potential and kinetic), and that under proper conditions the one can be converted into the other in such a way the sum of the two remains unchanged (conservation of energy).

Perhaps not surprisingly, the jumping off point is again found in a discussion attributed to Galileo. He was concerned about ways of analyzing motion. As above, where he used an inclined plane to slow motion, he thought of ways in which various qualities of motion could be measured. How far up a plane would a ball roll? How was this related to its speed at the bottom? How was this related to the height from which it rolled? Or to the incline? We can easily answer these questions in the abstract, idealized case, but the ugly bugbear of friction gets in the way of too many attempts to demonstrate these ideas. No wonder the Greeks avoided experiments! Nevertheless, Galileo did come up with an experimental device which is worth pursuing in this context. It is the pendulum.

These experiments are typically conducted as group discussion-participation sessions, not in small groups or individually. The reasoning between experiments is too close to be left to chance.

If a pendulum on a fairly long string is set up to swing parallel to a blackboard there are several observations that can be made fairly easily. If students push the pendulum from the rest position with different speeds it is seen that the height to which it rises increases with the speed. Analogously, if it is released from different heights it is obvious that it passes through its rest position at different speeds. And, if the height from which it is released is marked, it is seen that it rises to essentially the same height.

Now, if a stop is placed directly below the support so that as the pendulum swings its string is essentially shortened when it hits the stop, the height to which it rises is again seen to be the same, even though the path it takes is clearly different. The conclusion that starts to emerge is that the height to which a pendulum rises is dependent on its speed at the bottom and not on its path to the top. By the same token, its speed at the bottom depends on its or