Determine a mathematical model which describes the height of a ball in terms of the number of bounces after it has been dropped from a given height.
If a ball that bounces, like a basketball, is dropped (not thrown) from a given height, then it will bounce to a height less than the previous height. Can you determine a mathematical equation that will give the height of the ball after a certain number of bounces?
What equation do you suspect will model this situation? Think about what factors affect how high the ball will bounce.
- Balls (ones that bounce, i.e., basketball, small rubber balls, etc.)
- A large sheet of paper (about 2 meters high)
- A hard surface floor
The procedure is as follows:
- Attach the large sheet of paper to the wall. Mark the paper in reasonable increments.
- Estimate where you think the top of the ball will return. Mark and label it.
- Drop the ball from the 2-meter mark (any height that is reasonably distanced from the ground). Be sure to record the height consistently: either always record the height at the top of the ball or bottom of the ball.
- Get a friend to watch the ball come back up and see if the basketball reaches the marker. For instance, if it goes past the marker, then mark again and relabel and so on.
- Make a record of all the data points with respect to the bounce number.
Take the data points and see if you can find any correlation between the data points. It might be a good idea to graph the data points in Excel and analyze the data.
Did the model you came up with depend on the factors that you thought it would? Try using other bouncy balls to confirm your results. What parameter in your equation determines the "bounciness" of the ball?
Can you think of other real-world problems that use this type of pattern?