In this section we shift gears a bit and look at falling objects. This is a great classroom and playground subject. We can do lots of experiments using the TI-CBL. It is also a good starting point for studying space flight. We develop a new tool -- Newton divided differences -- for studying the motion of falling objects.

This is another very rich subject that can be used in high school algebra or calculus classes or in high school physics classes. Some of the material in this module can also be used with much younger students to illustrate the power of simple subtraction and division.

Two forces act on a falling object -- the force of gravity and the force due to air resistance. The relative importance of these two forces depends on the density and shape of the object and on its velocity. For a dense streamlined object like a baseball, the force of gravity is much more important than the force due to air resistance unless the object is going very fast. For an object with very low density, like a balloon, or an object that is moderately dense and not very streamlined, like a pillow, air resistance becomes important at even modest speeds. We begin by using the TI-CBL to collect some data.

You may want to look at the module on units and force and acceleration before continuing with this module.

The picture below shows the TI-CBL connected to a TI-82 and the "motion detector." The first thing to know about the motion detector is that it is misnamed. It is actually a range finder -- the same unit used on Polaroid cameras to measure the distance from the camera to the subject. It works by timing how long it takes a sound wave to make the round trip from the detector to a target and back.

There seem to be two philosophies about using the motion detector to record the travels of a falling object. They are illustrated in the figure below.

• Some people (including the author) hold the CBL high off the ground pointing downward. I usually stand on a chair. Then they hold the about-to-fall object 1.5 feet directly below the CBL and drop it, recording data as the object falls away from the CBL toward the ground.

• Other people place the CBL on the ground pointing upward and hold the about-to-fall object directly above the CBL. They drop the object and record data as the object falls toward the floor and the CBL. They catch the object before it hits the CBL, usually about 1.5 feet above the CBL. This author prefers the first method to avoid endangering the CBL.

Assemble a collection of objects that you want to try. Pillows are good. Books are good. A plastic grocery bag stuffed with crumpled newspaper is good. Be inventive. You want some dense objects for which gravity will be the only important force and you want some objects whose density is so low that air resistance becomes the most important force over the short distances we will use. Racquetballs are very good if you'd like to record data for a bouncing ball but because of their small size you may have to try several times before you gat a good data set.

The buttons below lead to instructions and programs for the TI graphing calculators for using the TI-CBL with the motion detector to record data about the travels of a falling object. When you click the appropriate button a new window will open with instructions for the calculator you chose. You should arrange these two windows so that it is easy to move back-and-forth between the them.

Your TI graphing calculator window has instructions for preparing the CBL, motion detector, and graphing calculator to record an experiment. Follow those instructions and then return here.

At this point your CBL, motion detector, and graphing calculator should still be connected and the CBL and graphing calculator should be turned on. Your motion detector is probably buzzing and the word READY should appear on the CBL screen. I recommend that you hold the TI-CBL high off the ground pointing straight downward and that when you run the experiment you drop the falling object so that it falls away from the motion detector but you may prefer the other approach because everything can rest on the ground. With three people the pointing down approach is not at all awkward.

Now hold the object steady and prepare to drop it. Have a friend press the CBL TRIGGER key and then immediately drop the object. If you are dropping it toward the CBL on the ground be sure that someone catches it before it hits and possibly damages the CBL.

The Graphing calculator will automatically recover the data as soon as the experiment is concluded. On the TI-92 the data will be in a list called data in the other graphing calculators it will be in the list L₁. You should do a series of experiments using different objects. To save the different experiments either move the data from each experiment to another graphing calculator by linking, or to a computer, or on the same calculator to another list. For example, I did an experiment dropping a pillow and then moved the data to a list called pillow as shown in the TI-92 screen below.

The data from this particular experiment is shown in the TI-92 screen below. Notice that this data was collected dropping the pillow toward the motion detector. At first the height is constant because the pillow is being held at a fixed height above the motion detector. Then as the pillow falls the graph descends. There is a sharp spike -- probably the pillow tumbled -- and after the spike the data is pretty useless. This is typical of real data involving household objects like pillows. We often get some measurements that aren't very useful because something unintended (like the tumbling of the pillow) occurred.

Gravity

The force produced by gravity causes an object to accelerate at a constant rate, g, measured in units like feet/second² or meters per second². As a result the velocity is a linear function

where t denotes time. The constant b is the object's initial velocity at time t = 0.

If an object is falling and the only force acting on it is gravity then its height is given by a function of the form

In the classroom you can exploit this situation to talk about coordinate systems in a natural setting. The data that you have collected is all measured in terms of feet from the motion detector. Thus, the location of the motion detector determines the origin of the coordinate system. If the motion detector is lying on the ground and pointing upward then a positive distance is height above the ground. In this situation the constant g is negative because the force of gravity is pointing downward. As the object falls under the influence of gravity the readings collected by the CBL decrease.

If the motion detector is held high in the air pointing downward then a positive reading is distance below the motion detector. In this situation as the object drops the readings recorded by the CBL increase. The constant g is positive.

Air Resistance

Air resistance is more complicated than gravity because the force due to air resistance depends on the shape of an object and on its velocity. For our purposes here the most important aspect of air resistance is that it increases as the velocity increases. There is a velocity at which the force produced by air resistance is equal to the force produced by gravity. This velocity is called the terminal velocity. Once a falling object is close to its terminal velocity, its acceleration is almost zero and, thus, its velocity is almost constant. This leads to an equation of the form

for the height, where v is the terminal velocity and the constant, b is determined by the initial conditions.

Newton Divided Differences

We are interested in two different kinds of falling objects.

• A falling object that is quite dense and reasonably streamlined, so that the only important force is the force of gravity. Thus, its height is given by an equation of the form
  

• An object that is not very dense and not very streamlined, so that its terminal velocity is very low. In this case after the object has fallen for a short time (and its velocity is very close to its terminal velocity) its height is given by a function of the form
  h(t) = v t + b

Thus, we need a technique for examining data and to see if it fits either of the two models above.

Our data consists of height measurements, h(t_i), taken at equally spaced points in time.

For example, we might make measurements beginning at time t₁ = 0, the time at which the object is released and then every 0.1 seconds -- at times

If q denotes the length of time between successive measurements, notice that for each i,

and

Now if our data can be described by the second (high air resistance) model and the object has been falling long enough so that its velocity is close to the terminal velocity then

We have

The quantities

on the left hand side of the equation above are called the first order Newton divided differences and are easily computed from the data. The computation above was based on one possible theoretical prediction of these quantities -- based on a high air resistance model. If the first order Newton divided differences computed from actual height measurements are all close to the same constant then the second (high air resistance) model does a good job of describing this data and the value of the constant is the terminal velocity. Remember that it takes a while for the falling object to get close to its terminal velocity.

If the first model

describes the data and we compute the first order Newton divided differences we obtain

and now if we compute the divided differences of the divided difference (these are called the second order divided differences) we obtain

Thus, we see that in theory if the first model does a good job of describing the data then the second order Newton divided differences will be constant and the value of the constant will be the constant acceleration, g, due to gravity. Similarly, in theory if the second model does a good job of describing the data then after a short time the first order Newton divided differences will be constant and the value of the constant will be the terminal velocity.

The bold face type used for the words in theory is very important. This is a place where theory often does not work in practice. The problem is that data often is not perfect and the data we are working with in this module is very likely to be far from perfect. As an object falls it may tumble a bit and the sound wave from the motion detector may bounce off different parts of the object. As a result the measurements we obtain usually have small errors and sometimes have large errors.

Even if the distance measurement are not too bad the first order divided differences, which are essentially velocity estimates, may have very large errors. The picture below illustrates why this happens.

This picture is hard to read and the reason it is hard to read is the reason that small errors in distance measurements can produce big errors in the first order Newton divided differences. The two vertical lines represent two successive times at which distance measurements were taken. The black line intersects these two lines at the actual error-free distances. The red line intersects these two lines at not-quite-perfect measurments of the distance. Notice the intersections of the red line with the two vertical lines are close to the intersections of the black line with the two vertical lines. The errors in the distance measurements are quite small. In fact, you can hardly see the error on the graph. The slope of the black line is the theoretical first order Newton divided difference -- that is, the Newton divided difference that we would compute if we had perfect data. The slope of the red line is the first order Newton divided difference that we would compute based on the actual data. Notice how far apart the two slopes are.

Another route to understanding this phenomenon goes by way of numeric computations. Suppose that we are working with two measurements taken 0.02 seconds apart and that the object is moving at a velocity of 5 feet per second. Thus, during the time between the two measurements it moves 0.1 foot. Suppose that the actual perfect distance measurements would be 6.00 feet and 6.10 feet. Now suppose that there is a small, 0.01 foot, error in each of the two measurements and the actual measurements are 6.01 feet and 6.09 feet. The computed velocity (or first order Newton divided difference) would be 4 feet per second. Two very small errors (roughly 0.2 %) in the distance measurements have lead to a huge (20%) error in the first order Newton divided difference.

• Suppose an object is moving at 5 feet per second and two successive distance measurements are made 0.05 seconds apart when the object is at a distance of roughly 6 feet. If the distance measurements have an error of at most 0.2% how far off can the first order Newton divided difference be?

• Suppose an object is moving at 5 feet per second and two successive distance measurements are made 0.10 seconds apart when the object is at a distance of roughly 6 feet. If the distance measurements have an error of at most 0.2% how far off can the first order Newton divided difference be?

• Suppose an object is moving at 5 feet per second and two successive distance measurements are made 0.20 seconds apart when the object is at a distance of roughly 6 feet. If the distance measurements have an error of at most 0.2% how far off can the first order Newton divided difference be?

Based on the exercises above you can see that one way to control these errors is by making fewer measurements with more time between the measurements.

This problem is even more serious for second order Newton divided differences because distance measurements with only a small error can result in modest errors in the first order Newton divided differences. But then these modest errors in the first order Newton divided differences can result in large errors in the second order Newton divided differences.

Do some very careful experiments with your CBL and see if you can get reasonably good results.