SPORTS The center field fence in Yankee Stadium is 7 feet...

SPORTS The center field fence in Yankee Stadium is 7 feet high and 408 feet from home plate. A baseball is hit at a point 3 feet above the ground. It leaves the bat at an angle of $\theta$ degrees with the horizontal at a speed of 100 miles per hour (see figure). (a) Write a set of parametric equations that model the path of the baseball. (b) Use a graphing utility to graph the path of the baseball when $\theta = 15^{\circ}$. Is the hit a home run? (c) Use the graphing utility to graph the path of the baseball when $\theta = 23^{\circ}$. Is the hit a home run? (d) Find the minimum angle required for the hit to be a home run.

SPORTS The center field fence in Yankee Stadium is 7 feet high and 408 feet from home plate. A baseball is hit at a point 3 feet above the ground. It leaves the bat at an angle of $\theta$ degrees with the horizontal at a speed of 100 miles per hour (see figure).

(a) Write a set of parametric equations that model the path of the baseball.
(b) Use a graphing utility to graph the path of the baseball when $\theta = 15^{\circ}$. Is the hit a home run?
(c) Use the graphing utility to graph the path of the baseball when $\theta = 23^{\circ}$. Is the hit a home run?
(d) Find the minimum angle required for the hit to be a home run.

Key Concepts

Projectile Motion

Projectile motion describes the motion of an object thrown into space, subject only to the acceleration of gravity. This concept uses Newton’s laws of motion and is typically analyzed through both horizontal and vertical components that are independent from each other, with the vertical motion subject to a constant acceleration due to gravity while the horizontal motion maintains a constant velocity (ignoring air resistance).

Parametric Equations

Parametric equations are used to model the trajectory of an object by expressing its horizontal and vertical displacements as separate functions of time. In the case of projectile motion, one equation typically represents the horizontal displacement (x = function of time) and the other represents the vertical displacement (y = function of time, including the effect of gravity), allowing for a complete description of the projectile’s path.

Initial Conditions and Conversion of Units

Initial conditions specify the starting position and velocity of the projectile. These conditions include the initial height, launch angle, and speed at which the projectile is hit or thrown. Often, it is necessary to convert between different unit systems (for example, from miles per hour to feet per second) to maintain consistency in calculations, especially when working in a specific unit system like feet and seconds.

Graphical Analysis of Trajectories

Graphing the path of a projectile using a graphing utility allows for visual analysis of the projectile's motion. This can help determine whether the projectile meets certain criteria, such as clearing a barrier or reaching a specific point. The graph visually represents the trajectory, making it easier to analyze the effects of different launch angles and speeds on the projectile's path.

Optimization in Projectile Motion

Optimization in the context of projectile motion involves finding the conditions (such as the minimum launch angle or the optimal initial speed) needed to achieve a particular objective, like clearing a fence or reaching a target. This typically requires analyzing the mathematical model of the trajectory and adjusting parameters to meet the desired outcome, often through a combination of algebraic manipulation and graphical techniques.

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