Background

Collecting and analyzing data is the most important thing a scientist will do in the physics lab. Indeed, 90% of what is done in any physics experiment is collecting and analyzing data. One of the most useful tools available for this analysis is the graph.

There are many benefits to plotting a graph, but in general, graphs allow someone

1. to visualize the data collected better

2. to see whether or not there is a trend in the data, and

3. to make predictions beyond what was tested experimentally.

The trend referenced above means a functional (mathematical) relationship between the variables tested. In this course, students will generally be confirming a relationship already known to exist between the variables.

Figure 1: Sample graph displaying data in which the experimenter adjusted the "Tromhus" to see how the "Platzx" were affected by it.

Figure 1 shows a typical graph. Remember that the dependent variable "Y" is on the vertical axis, and the independent variable "X" is on the horizontal axis. By definition, the independent variable is the variable which is controlled (changes) in the experiment. The dependent variable is the one which is measured in response to these changes in the independent variable. Four data points are plotted, and a best-fit straight line has been drawn through them to show the (in this case, linear) trend in the data.

The slope of the line (or any linear line, actually)

is found by the formula: =

is constant

tells a reader how the dependent variable Y changes in relation to the independent variable X. If this ratio is greater than 1, it means that the dependent variable is changing faster (at a greater rate) than the independent variable. Conversely, a ratio less than 1 means that the independent variable is changing faster than the dependent variable.

This simple experiment will introduce a student to the concepts discussed above. A student will analyze collected graphically.

Apparatus

2 Bouncing Balls of different elasticity

2 Meter Sticks

Procedure A

The following part of the procedure (steps 1-3) is if you plan to collect your own data from materials at home. If not, I would recommend still reading through these procedures to understand how the data was taken, but you will not be completing the procedure.

1. Hold the 2 meter sticks vertically. Consider taping them to a wall so that no one needs to hold them throughout the experiment. When in proper position, one should be above the other on wall with the bottom end of the top one touching the top end of the bottom one. The bottom meter stick should be touching the ground.

2. Select a ball and decide on five heights from which a member of the group is going to drop the ball. These heights should be over as much of the range as is available. For example, 50cm, 100cm, and 150cm is better than 70cm, 80cm, and 90cm. [The heights should be measured from the bottom of the ball.]

3. Record the drop heights in Table 1. Drop the ball five times from each of these heights, each time recording the corresponding bounce height. It is helpful to either have a camera record the maximum height that the ball bounces to each time.

4. Calculate the average bounce height at each of the drop heights.

5. *Plot a graph (Bounce Height vs. Drop Height) to assist the group in interpreting the data. The AVERAGE bounce height will be graphed- not each value.

a) Scale both axes from 0 to a number larger than what was measured for that quantity in appropriate units.

b) Label the axes. The variables to be plotted are drop height and average bounce height. [Make sure that the plot has the correct variable on the correct axis. It matters. Don't forget the proper labeling of units of measure, proper marking of the axes, ....]

c) Add a trend line (line of best fit) to the graph corresponding to the five data points. (With five data points, the trend line should be parallel to an imaginary line which connects the two farthest points, but the trend line will only touch any of the data points, if the five data points are perfectly in a line.)

d) NOTE: If your graph does not look linear, check each of your data points to make sure that everything was inputted correctly.

6. *Determine the slope of this line and record the value. The slope is calculated by the computer and will always be the value multiplied by the independent variable (the "x" value).

Procedure B

7. Exchange the ball which the group used for a second ball. This ball should have a noticeably different bounce from the first ball. Repeat Procedure A with this second ball, recording the data in Table 2. [When finished, hang onto the ball; do not return it until the end of the lab.] Note that when performing steps 5 and 6 for the second ball, plot the data for the second ball on the same graph with the data from the first ball. Be sure to include a legend on the graph so as to differentiate the data.

Procedure C

8. *As stated in the Background section, graphs allow a student to make predictions beyond what was tested experimentally. Using the graph's trendline and equation, predict how high the first and second ball will bounce if dropped from a sixth height (one not already tested). To do this, you will plug your new drop height into your line of best fit equation and see what value is predicted by the equation. This will be the predicted bounce height for your given equation.

9. Actually drop both balls from this sixth height and compare the results to the prediction in step 8 (calculate

a percent error between the values).

Data and Results

Data and Results A

Table 1: First Ball Data

Drop Height (cm)

Bounce Height (cm)

Average (cm)

Data and Results B

Table 2: Second Ball Data

Drop Height (cm)

Bounce Height (cm)

Average (cm)

Data and Results C

Ball 1

Drop height (cm) ____________________

Calculated Bounce Height from trendline (cm) ____________________

Actual Bounce Height (cm) ____________________

% Error ____________________ (show work)

Ball 2

Drop height (cm) ____________________

Calculated Bounce Height from trendline (cm) ____________________

Actual Bounce Height (cm) ____________________

% Error ____________________ (show work)

Analysis A

1. Why is it better to use more of the range? In other words, why is it better to drop the ball from heights of 50cm, 100cm, and 150cm than from 70cm, 80cm, and 90cm? Be specific- because it is more accurate is not a sufficient response.

2. Measuring the drop height, why is it better to measure from the bottom of the ball and not the top or the center?

3. What is the slope of the trend line for the data of the first ball? What are its units- does it have any?

a. Equation of the line

b. Slope

c. Units (be careful: think about the equation for slope (located on the first page) and how this would affect the units.

Analysis B

4. When testing with the second ball, is it NECESSARY to use the same drop heights that were used for the first ball? Why or why not?

5. What is the slope of the trend line for the data of the second ball? What are its units- does it have any?

a. Equation of the line

b. Slope

c. Units

Analysis: Comparison

Let's call the slope of each of the plotted lines the bounce coefficient for that particular ball.

6. Describe how the two bounce coefficients compare to each other. How are they the same/different?

7. Does a larger coefficient mean that a ball is going to bounce higher or lower than a ball with a smaller coefficient?

8. A. What would a bounce coefficient of 1 imply physically (not mathematically)?

B. Would it be possible for a ball to have a coefficient greater than 1? Why or why not?

Analysis C

9. How successful was the prediction? Did your lines hit each one of your points exactly? What could be some possible sources of error? (Human error is NOT a sufficient response) What could you have done differently/what further experiments could you do to eliminate the error?

10. Will the trend shown on the graph between bounce height and drop height be accurate (or somewhat accurate) indefinitely, or will it fail at some point? This question is asking about the physical relationship between the drop height and the bounce height, not the mathematical relationship. Would it be accurate if you dropped your ball off of the Empire State Building? Why or why not?

11. What was the purpose of this lab? Did you achieve that purpose?