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Describing One-Dimensional Motion

Preliminary Version

Michael Fowler

Physics 581

Prelesson lab

Use of sonic ranger to measure position and motion – one dimensional movement only. The purpose of this lab is to learn about graphs of position against time, and of velocity against time. The aim is to reach a point where a student presented with a position/time or (one-dimensional) velocity/time graph will have some immediate understanding of the motion described. One standard approach is to have a position/time graph on the screen, which one student is looking at. Another student moves back and forth along a line with the sonic ranger at one end, tracking the student’s movements and displaying them on the screen. The student who watching the screen shouts orders to the student on the line, the aim being to reproduce the preexisting position/time graph. The students then trade places, etc.

Presentation Technique

In the material below, I intersperse statements with large numbers of questions and answers. The idea is that in presenting this material to a class, the questions could be used for class discussion. My basic technique here is to ask the students to think about the question, decide on a tentative answer, then discuss it with a neighbor, or possibly a larger group, say four. I would then ask for an answer from one of the groups. This doesn’t put individual students on the spot. Of course, some of the questions below are less suitable than others for extended treatment of this type, and perhaps in some cases the material should be just presented—it depends on the dynamics of the particular class.

The first recorded attempts at describing motion were made by the Greeks – but note that for the moment, we’re excluding motion of stars, planets etc., which had been described much earlier by Babylonians. We’re just looking at things moving here on earth.

The Greeks divided motion into two types: "Natural Motion" and "Violent Motion" .

Question: What do you think they meant by these terms? (Discuss)

Answer: natural motion was something "finding its proper place" – a stone in air would fall down, fire would rise. The Greeks believed everything to be made up of four elements: earth, water, air and fire, and when free to move an object tended to its proper place in this succession, earth at the bottom, then water, then air – with fire at the top.

(Most things were a mixture of elements, so it wasn’t completely obvious what their "proper place" was. What about wood, for example?)

Violent motion was forced motion – violence just means force. So, if I lift a stone, that’s violent motion – forced motion, the stone doesn’t want to go up!

Aristotle's Laws of Motion

Aristotle was the first to think quantitatively about the speeds involved in these movements – at least, the first whose work still survives. He made two quantitative assertions about how things fall (natural motion):

1.Heavier things fall faster, the speed being proportional to the weight.

2.The speed of fall of a given object depends inversely on the density of the medium it is falling through, so, for example, the same body will fall twice as fast through a medium of half the density.

Question: How do you think he came up with these rules?

Possible answers: The motion of, say, a falling stone, is hard to observe – it’s over so quickly. To digest what’s going on, it is necessary to slow down the motion somehow. Aristotle slowed down the falling motion by observing objects falling through water.

Question: What’s wrong with that? Falling through water does slow down the fall, making it easier to observe, right?

Answer: Yes, and Aristotle wasn’t far wrong about heavier things falling proportionately faster, after you allow for buoyancy (which we’ll discuss later in this course).

Question: But is the motion changed in any essential way other than just being slowed down? Is Aristotle missing something?

Answer: Yes – and apparently this didn’t occur to Aristotle, but it did to later Greeks, and, especially—much later—Galileo made it very clear. What is missing in Aristotle’s laws is the thought that a falling object might not just fall at one speed – in fact, if you just drop a ball or a brick, it picks up speed.

Question: But it’s all over so quickly – how could Galileo be so sure it was picking up speed? How can we see that without using modern technology?

Answer: Actually, Greeks after Aristotle remarked that water drops coming out of a spout were further apart after they’d fallen some distance than when they just left the spout, so the drops evidently picked up speed while falling. But Galileo had a much more convincing argument – if you drop a stone onto a stake which is pushed into soft earth, imagine how the further distance the stake goes in depends on how high the stone was before you dropped it on to the stake. If you’re not sure about that, imagine dropping a brick onto your foot (in shoes!) from a quarter of an inch, an inch, three inches, a foot, three feet, etc. you will suddenly realize that you already know things pick up speed as they fall!

Well, this perhaps makes it convincing that indeed speed increases as a brick falls, but physics is a quantitative science, so we need some quantitative statement about how the speed increases—and this is where Galileo made a great contribution to science.

Galileo’s Idea

Galileo made the simplest possible suggestion: speed increases linearly with time – he called it "naturally accelerated motion". "Acceleration" means adding speed "celer" is the root meaning speed.

Question: How could he check this hypothesis?

Answer: (sort of) He slowed down the motion enough to see what was going on by rolling a ball down a gentle slope rather than just dropping it.

Question: BUT—wait a minute—if dropping a ball through water changed the nature of the motion—changing it from constant acceleration to pretty much a steady speed—why should we believe that rolling a ball down a slope will not change the nature of the motion, but just slow it down?

Answer: Actually this is a pretty tricky one. Galileo thought about it a lot, and gave an argument based on observing a pendulum. The details are given in my lecture here.

(It’s not difficult to set up a demo of this, and it’s worth doing.)

(One point that is probably not worth bringing up right now is that Galileo missed something here—actually a ball rolling down a plane doesn’t reach the same speed as a falling ball at the corresponding height, but a block sliding down with negligible friction would—the difference being that the rotating motion of the ball takes up some of the energy.)

Establishing that the Acceleration is Constant

Of course, Galileo couldn’t monitor the increase of speed as the ball rolled down the slope. His best timing device was a water clock, turning a small spigot on and off and assuming the weight of water flowing in gives a direct measure of the time period. This will be completely unreliable for times of order a second, because of the reaction time of whoever turns the spigot, so it’s difficult to measure speed.

However, it is possible to measure times of a few seconds. Galileo’s strategy was to measure how long the ball took to roll all the way down the ramp (about ten seconds for the length and slope he used) then measure the time for some fraction of a distance, and compare the times.

Do Galileo’s Experiment!

What Galileo found was that in twice the time, the ball rolled four times the distance. He deduced from this that the ball gained speed at a constant rate. We’ll see later just how he deduced that, at this point we just want to show how he did the experiment. It isn’t that difficult, everybody can understand what’s going on, and it’s fun—several students can take part.

You need a ramp at least two meters long, say a 2x6 with a groove cut on the narrow side, parallel to the edge. (You could get away with something smaller, but this is about the size Galileo used, and it’s more dramatic.) The groove must be smooth to minimize friction. A pool ball or, even better, a one-inch steel ball is fine, it needs to be a fairly heavy ball to reduce the relative importance of friction. Let water flow out of a container into a polystyrene cup, weighed before and after, so you need a scale accurate to grams.

Take several readings at each distance. Try distances differing by a factor of four, but then try something different. Make up a table on the blackboard of times and distances.

Draw Graphs

First, everybody should draw a graph of position against time from the data gathered above. Admittedly this will be very rough, with perhaps three or four points (it goes through 0,0 of course) but it’s a start.

Next, everybody should draw a theoretical graph of velocity against time based on Galileo’s assumption that the ball picks up speed at a constant rate. Make some reasonable guess about how much speed it’s picked up in one second.

The next—crucial—exercise is to understand that the area under the velocity/time graph between two times corresponds to the distance traveled between those times. This idea should be built up by considering first a small interval of time during which the change in speed is negligible.

Once the area—distance moved relationship is fully appreciated, it is easy to see from the straight-line velocity/time graph how the distance goes as the square of the time.

Note: these ideas are thoroughly discussed here. (Check this page out before reading on—it also describes the video experiment below.)

Formulas

Only at this point should the formula for distance fallen in constant acceleration, s = ˝gt2, be introduced, and illustrated by looking at the area under the velocity/time graph. Another velocity/time graph can be drawn for nonzero initial velocity, first with initial velocity downward then upward, so that the formula s = vt + ˝gt2 can be understood for both cases from the graph. These simple graphs are perhaps better drawn without using a spreadsheet, but on graph paper, so the students associate the formulas with the pictures. It is also worth drawing position/time graphs illustrating these cases.