Question

Consider a projectile launched at a height \(h\) feet above the ground and at an angle \(\theta\) with the horizontal. If the initial velocity is \(v_{0}\) feet per second, the path of the projectile is modeled by the parametric equations \(x=\left(v_{0} \cos \theta\right) t\) and \(y=h+\left(v_{0} \sin \theta\right) t-16 t^{2}\) The center field fence in a ballpark is 10 feet high and 400 feet from home plate. The ball is hit 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 for the path of the ball. (b) Use a graphing utility to graph the path of the ball when \(\theta=15^{\circ} .\) Is the hit a home run? (c) Use a graphing utility to graph the path of the ball when \(\theta=23^{\circ} .\) Is the hit a home run? (d) Find the minimum angle at which the ball must leave the bat in order for the hit to be a home run.

Short answer

One would use the parametric equations to analyze the path of the baseball for different initial angles of projection. The minimum angle at which the ball must leave the bat in order for the hit to be a home run varies and needs to be determined graphically. It is clear whether or not a hit is a home run from the graphs plotted at specific angles.

Step-by-step solution

Write Parametric Equations for the path of the ball

Given that the initial speed of the ball is 100 miles per hour, this speed must be converted to feet per second (multiply by 5280/3600 to convert). The height from which the ball is hit is 3 feet. Using these data, we can write the following parametric equations: \[x=(v_{0} \cos \theta) t\]\[y=h+(v_{0} \sin \theta) t-16t^{2}\]Where: \(v_{0}\) is approximately 146.667 feet per second, \(h\) = 3 feet, and \(\theta\) is the angle at which the ball is hit.

Graphing the path of the ball at \(\theta = 15^{\circ}\)

By using a graphing utility, substitute \(\theta\) in the equations obtained in step 1 with 15 degrees. As shown by the graph, the path of the ball's motion can be traced. If the ball goes beyond 400 feet in distance and remains above the 10-feet fence, it is considered a home run.

Graphing the path of the ball at \(\theta = 23^{\circ}\)

Similarly, following step 2, replace the angle in the parametric equations with 23 degrees. If the ball in the graph goes beyond 400 feet in distance and remains above the 10-feet fence, it is considered a home run.

Find the minimum angle for a home run

To find the minimum angle at which a homerun is possible, we need to find the minimum value of \(\theta\) for which the ball crosses the 400 feet mark while maintaining an altitude of at least 10 feet. To solve this, iterate over different values of \(\theta\) using the graphing utility until the minimum value is found