Lisa Petraitis, Jeff Walraven
In our excel problem students will study the motion of a basketball. The point of the problem is to find the best angle or the best velocity the ball should be thrown so that it will go in the basket from the free throw line.
The goals of this activity in connection to the standards would be to select and use appropriate technology, instruments and formulas to solve problems, interpret results and communicate findings. Depending on how much information the teacher wanted to give to the students this problem could be used at the middle school level or the high school level.
We decided to approach this problem by drawing an appropriate picture. Then we constructed a table to see which values we knew and which ones we needed an equation for. Next we decided that there were three unknown values. We needed to choose two to stay constant and one that could be dependent upon other variables. We decided velocity and the angle (alpha) the ball is thrown should be kept constant. By using simple physics we got the equations for the velocity vectors. Velocity in the x direction equals velocity times cosine of alpha. Velocity in the y direction equals velocity times the sine of alpha. We decided time should be 0, .1, .2, .3,... etc. The motion of the basketball in the x direction equals velocity in x direction times time. The motion in the y direction equals velocity in the y direction times time minus gravity divided by two times time squared. Plotting the x direction and y direction on an xy-plane gives us a picture of the motion of the basketball.
If we give the formulas for the velocity vectors and the x and y directions of the basketball then this problem would be appropriate for any level of algebra students. This will help them learn equations with variables and representations of these equations. We could also give this problem to calculus students without the equations for the velocity vectors. Then they can learn about directional vectors as well as representations of this motion. Also, as a break from the routine class the teacher could bring the class down to the gym and videotape them shooting a free throw. The teacher could also bring in a videotape of a NBA or NCAA game and they can observe the parabolic motion of a basketball.
From doing this problem we found the best or most realistic velocity would be 7.6 meters per second. We also found that the best angle would 60 degrees. These two values would give the best graph for sinking a free throw shot.
Excel is extremely beneficial in this problem because students can easily change the values of certain variables and see the results within seconds. They could try keeping different variables constant. By easily changing variables they will be able to find the best graph for the motion of a free throw shot. Once the numbers are changed on the table the graph automatically changes. Doing this with paper and pencil will take much longer to find the correct motion.
How a Basketball Flies
Do a simulation of a shot from the free throw line. The free throw line is 4.60 m from the center of the basket. The basket has a diameter of 0.45 cm, its height is 3.05 m. The ball has a diameter of 24.6 cm.
1. Give the player some hints how to throw the ball. Think about the velocity of the ball, if it leaves the hand of the player. Think about the angle - related to the horizontal - it should be released. Think also about the angle of entry into the basket.
2. Take a videotape of shot from the free throw line and compare it with your results.
Solution (EXCEL FILE)
(Lisa Petraitis, Jeff Walraven)