Quadratic Function Project Part 3: Parabolic Trajectories
in Sports
For those of you who don’t understand the rules of Major League Baseball (MLB), there are two places where a batter can stand to hit the ball. They can stand on the right side (Right Handed Batter) or on the Left Side (Left handed Batter). Pitchers throw the baseball in order for this ball to go through the strike zone (see image right) without the batter making contact. This is called a strike. The Image to the right is from the prospective of the pitcher. The pitcher has to throw a pitch that is about 2.3 to 3 feet above the ground for it to be in the “strike zone”
1) Orlando Hernandez although known for his fast pitches, often times threw pitches at different speeds. The trajectories below represent two different pitches, thrown at different speeds. The top trajectory graph is a pitch thrown at 50 mph called for a strike. The bottom trajectory graph is a pitch thrown at 48 mph called for a ball (a ball is a pitch that does not go through the strike zone, refer to images above).
a. Using , where is the initial velocity and is the initial height. Write an equation for both of these trajectories
b. How much time does it take for the 50 mph pitch reach the batter?
c. How much time does it take the 48 mph pitch to reach the batter?
d. What is the difference in the height of the pitches when the ball crosses the home plate? What does this mean?
e. How far behind the batter would the 50 mph pitch land if the catcher does not catch the ball?
f. From what height does “El Duque” need to release his 48 mph pitch such that the end point is the same as the 50 mph pitch? Is this a reasonable scenario? Please explain your response.
g. Two different pitchers learned to duplicate these pitches perfectly. They stood 40 feet apart, facing each other, and threw directly at each other. The pitcher on the left threw the 50 mph pitch and the pitcher on the right threw the 48 mph pitch. When and where would these two pitches collide?
2) The picture below represents the trajectory of three different basketballs thrown by the same person from the same distance away from the hoop. The only thing different between each thrown is the angle at which the shooter decides to release the ball. Notice that the shooter is releasing the ball at a different angle, but the ball still makes it into the hoop. Top trajectory is the 53-degree shot, middle trajectory is the 45-degree shot and the bottom trajectory is the 35-degree trajectory shot.
a. If the shooter releases the ball at (0, 7) and all the balls go through the hoop at the point (13.5, 10), using the other points provided, find the equations for all three trajectories.
b. Identify the maximum height the ball reaches at each of the different three trajectories
c. The front of the rim is located at (13, 10). How far back does the shooter need to step back for the ball to bounce off the front of the rim?
d. Chose one of the three trajectories. From what height does the player need to release the ball (still 14.5 feet from the backboard), such the ball bounces off the top of the backboard at (14.5, 13), where the shots continue to have the same trajectory as before?
e. During half time a child wins a chance to shoot the basketball with the team. The child tries to get as close as possible to the hoop to increase his chance of making the shot. The child stands 10 feet from our shooter. The child proceeds to shoot the ball “granny style” and releases it 4 feet from the ground. If the ball must make it through the hoop at (13.5, 10), what is the maximum height that the ball reached in order to make it into the hoop?
i. If the player where to take his three shots again, at what point would the child’s ball collide with each of his three shots?