How fast can you throw a tennis ball?

The height in feet of a ball thrown directly up at any time t can be approximated by the following formula:

where t is time in seconds, is the initial velocity in feet per second and is the initial height in feet.

You are going to need:

  • tennis ball
  • stop watch
  • tape measure

It is important that the ball is thrown directly up.

Step 1: Measure how high the release point is for the thrower, this is .

Step 2: Throw the ball directly up and time it from when it leaves the throwers hand until it hits the ground. (The higher you throw it the better your results).

Step 3: Repeat in order to have 3 trials. (one for each member of the group)

You need only keep the "best" time, or alternatively, average the times for one thrower. (The release point should remain the same) Your goal is 3 seconds.

Questions:

  1. When the height is zero, what time, t, is this corresponding to? [ f(t)=0] (Where is the ball?)
  2. You should know everything in the equation for when height is zero. This will allow us to solve for the unknown. Determine the general equation for f(t) by solving for . (It is important that the units for are in feet)
  3. Just for fun, what is your initial velocity in miles per hour?______
  4. Open the following Sketchpad file. Tennis ball project This file lets you change the value of and , the units are ft/sec and feet respectively. Simple double click the values and enter the value you calculated for the experiment.
  5. At what time does the ball reach its highest point? Be sure you give units.(Solve algebraically)
  6. How high does the ball get? Be sure you give units (Solve algebraically)
  7. Label all relative information. Max/Min, X/Y intercepts etc. (This is a function of t). You can use the Graphing Toolkit options in Sketchpad to put them on your graph [Click on and then select graphing toolkit and then the option you desire],
    Take a screenshot of the graph in Geometer's Sketchpad and insert below. Be sure that you label the units for each axis.
  1. Have Geometer's Sketchpad create a line that goes through points A and B. Simply select each point and the select Construct | Line. Measure the slope of this line. With the line selected, choose Measure | Slope. Record the value that Sketchpad produces. ______
  2. Looking at the graph in Geometer's Sketchpad, and the labels you put on the original graph:
  3. What units should you include with your last answer? ______
    As a hint press the button
  4. Justify your answer. ______
  5. What meaning does this answer have? Is there a more common way that it is referred to?
    ______
  6. Given your function for the height of the tennis ball at any time:
  7. calculate the difference quotient

  8. Evaluate the difference quotient if x =1 and h =2. What is the meaning of this answer? Be sure to give units. (Hint: Look at points A and B as they are right now as well as your calculation above.)
  9. The value calculated above in part b, is really just what idea graphically? We should have seen this answer before. Be specific.
    ______
  10. Look back in Geometer's Sketchpad and write a description of what h is.
    ______
  11. Double click on the value of xb. Change its value to correspond to an h value of 0.5 for the difference quotient. (Note xb ≠ 0.5) Use a value that is shown in Geometer's Sketchpad to calculate:

Verify your answer by using these values in your difference quotient calculated in 9a.

  1. Summarize what your conclusion is about what the difference quotient tells us about throwing the tennis ball in the air. Be sure that you state what x and h are and how they relate to your conclusion. As a hint, think about the units again. Be as specific as possible.
    ______
  2. If h continues to get smaller, what does this mean graphically? Press Describe what you notice
    ______
  3. If h is zero, what kind of line should Sketchpad draw at point A?______
  4. Going back to the difference quotient you calculated in 9a, substitute the value of 1 in for x and the value of 0 in for h. What is the value of the difference quotient? Be sure to include units. ______
  1. Press . With the line selected, choose Measure | Slope. Record the value that Sketchpad produces. ______
  1. What meaning does this slope have? How does this relate to what you wrote in 12b

______

  1. Summarize your understanding of how this line relates to the difference quotient and to the tennis ball.
  1. Based upon everything, explain how to find the velocity of the tennis ball after it had been in the air for three seconds. Explain it by looking back at the difference quotient. Also, double click on xa Change the value to 3. How does this confirm your result?
  2. Will the velocity of the ball ever be zero? Explain.

______

  1. Summarized what you have learned from throwing a tennis ball into the air. ______

© Dr. Kevin A. Thompson