Question

Parametric Equations Project (Launch Angle)

A 90mph fastball can leave a player’s bat at the speed of 110mph. Suppose a
batter hits such as fast ball at a height of three feet above the ground to dead
center field where the distance of the 12 foot high outfield fence is 410 feet
from home plate. The “launch angle” is the vertical angle at which a ball leaves
a player’s bat after being hit.
Assuming the only force acting on the ball is gravity (−?? ??/?
?
), determine
whether the batter hit a home run if the launch angle is ??°. Round all
calculations to three decimal places.

Submission: You will be graded on both your math and writing skills.
Therefore, all your written responses should be given in complete sentences, and
all mathematical expressions should be well-defined and include proper notation.
MS Word provides an equation feature under “Insert” which will allow you to
format the mathematical expressions.
Your submission must include the title of the project and underneath it your name.
The project will have three sections that must be titled “Approach”, “Solution”
and “Reflections”. The format and grading of these sections is as follows.
Approach: (5 points)
Write two or three paragraphs describing your approach to solving this problem.
This should include the steps you will take as well as how you will determine
whether your solution is reasonable.
Solution: (30 points)
In this section, you will provide your mathematical solution to the problem. Each
mathematical expression you provide as part of your solution must be described
or explained in writing. The manner in which your textbook presents solutions to
word problems is a good model for this section. Be sure to provide a sufficient
number of steps in detailing your solution. The textbook often skips step in order
to limit page count; you are not under such a restriction; so don’t skip steps. It
will count against you.
Reflections: (5 points)
Write a paragraph or two reflecting on this assignment. What did you learn from
the assignment? Were there any calculations that were surprising to you? What
changes might make the project more realistic?

Answer 1

The following figure shows a rough sketch of the projectile

The given parameters are

Launching angle

Horizontal distance to be covered

Launching Height and height of the fence to be overcome by the ball at distance is

Acceleration due to gravity

The horizontal component of the velocity

The vertical component of the velocity

To hit a home run the height of the ball should be greater than the 12 feet at distance 410 feet from the home plate.

Using kinematic equations for projectile motion

For horizontal motion

As there is no acceleration along x axis

.................................................................................................................(1)

For vertical motion

.................................................................................................(2)

Let us first check that the projectile will cover a horizontal distance or not, by using formula of range of projectile

Thus with these launching angle and launching speed ball will cover the outfield distance.

Now to find out what is the height of the projectile when its horizontal distance is 410 feet

For that first calculate time require to reach 410 feet by using equation (1)

Substituting t=2.65 sec in equation (2) to determine the height of the projectile at distance 410 feet from the home plate.

Thus the projectile is 15.614 feet above the ground when it is at distance 410 feet from the home plate, clearing the height of the fence.

Hence batter will hit home run with the launching angle and launching velocity .