Coefficient of Restitution

Fortran Exercise #2: Coefficient Of Restitution

1. DEFINITION

In this lab, we shall be studying the coefficient of restitution of a superball. The coefficient of restitution, COR, is the ratio of the bounce-back velocity to the original velocity of an object undergoing impact (such as a ball impacting the ground after being dropped from an initial height). Using the principle of the conservation of energy, the COR can be used to relate the bounce-back height to the original height of a ball dropped from rest. In fact, this ratio is equal to the square of the COR. Mathematically, if a ball is dropped from a height h(0), and bounces back to a height h(1), then:

As the object hits the ground, it is compressed (like a spring) such that the energy of the ball is transferred into the compression of the material. Some of this energy is converted to heat, and lost. The remainder is available to launch the object back up into space after the bounce. The fact that the ball bounces back to less than its original height indicates that the maximum value of COR is 1, and values progressively less than this indicate that less and less of the original energy is available after the bounce.

The COR is a mechanical property of a specific impact, and involves the material and geometries of both bodies. Consider a superball being bounced upon a concrete floor. A superball (made of superball plastic) bouncing upon the concrete floor has a different coefficient of restitution than a sphere of the same size made of rubber or of another plastic bouncing upon the same concrete floor. Conversely, a superball would bounce back a different amount from a concrete floor than if it bounced upon a flat surface of mud (as an extreme example). Also, the coefficient of restitution of a rubber sphere is quite likely different from that of a rubber cube dropped upon the same surface. So, the COR is dependent upon the geometries of the two objects undergoing impact and their compositions (materials).

2. PROCEDURE

In this exercise, we shall begin by doing an experiment. We shall drop a superball upon the floor from an initial height, h(0), of one meter. After the first bounce, we shall call the height above the floor h(1); after the second bounce, h(2); after the ith bounce, h(i). Thus, the COR for each bounce may be defined as:

Notice that COR is a ratio of heights and as such is unitless.

This will allow us to fill out the following table:

i h(i) COR(i)

(m)

--------

0 1. -

1 - -

2 - -

3 - -

4 - -

5 - -

Because we are using a meter stick and estimating the heights of the bounces by eyeball, we should take about five readings to obtain a good average set of heights (hoping that the errors will average out over the five readings).

We will generate code to:

Input the measured values of bounce-back heights

Input a (guess for the) coefficient of restitution, COR

Compute, from the COR input, the calculated bounce-back heights

and

Compare the results

You will be given a program which includes most of the instructions to solve the problem. Your task will be to fill in the missing instructions and data. Specifically, you must: (1) enter the 5 measured bounce-back heights, (2) prompt for, read in, and echo an average (average over all five bounces), coefficient of restitution and the initial height of the ball, (3) calculate the bounce-back heights from the initial height and an average COR, and (4) after each calculation, print the bounce number, the calculated bounce-back height, and the measured bounce-back height on the same line. As you run the program, you will enter a guess for the COR and the initial height of the ball. The program will then compute and print out the calculated bounce-back heights. Compare the calculated heights to those measured at each bounce, and determine a better COR for the experiment. Keep running the program using different CORs until you obtain a "best" COR (to do this, you will have to keep running the program with different values of COR until you find one where the 5 calculated heights most closely match the measured bounce-back heights.)

Here, we define the variables for you, as follows - bounce number = i (integer)

2.1 Input quantities (to be echoed):

initial height = h(0) (real)

coefficient of restitution = COR (real)

2.2 Calculated quantities:

calculated height of first bounce = h1c (real)

calculated height of second bounce = h2c (real)

etc.

calculated height of fifth bounce = h5c (real)

2.3 Measured quantities:

measured height of first bounce = h1m (real)

...