Project

ENGR 2120: Dynamics Project #2a

Project Assignment For this part of the project assignment, you will be designing a machine that is capable of serving a tennis ball! The motivation behind building such a device is to allow a tennis player to continuously work on his or her return of serve game.

Figure 1: Tennis court schematic

Imagine you are an engineer working for a tennis company, and you are tasked with determining certain design specifications for the tennis ball launcher shown in Fig. 1. The design parameters are:

𝑡𝑡, the time of flight of the ball (in seconds)

𝑦𝑦0, the initial height of the ball (in meters) 𝜃𝜃, the fixed angle of the tennis ball launcher (in degrees) 𝑟𝑟,̇ the initial speed of the ball (in m/s)

The design parameters must be determined that will allow for the following design targets:

𝑑𝑑 = 15 m, the distance in the 𝑥𝑥 direction 𝛼𝛼 = 16∘, the angle between the 𝑥𝑥-axis and the final velocity vector 𝑣𝑣 = 31 m

s , the speed of the ball when it hits the tennis court

Please follow the steps below to accomplish your task:

1. Include in your report a general discussion of projectile motion (you may want to make this a new section or subsection). For example, you may discuss the following:

a. How does one know when to use projectile motion equations?

b. When are the projectile motion equations valid? c. Why do you typically use rectangular coordinates when dealing with projectile

motion? 2. Determine the time of flight of the ball, 𝑡𝑡. That is, how long does it take for the ball to strike

the ground at position 𝑑𝑑? a. Please show the derivation in variable form. That is, your final answer may only be a

function of some or all of the following parameters (𝛼𝛼, 𝑑𝑑, 𝑣𝑣, 𝑔𝑔). b. Please provide the numeric answer based on your equation in part 2a. Please be sure

to include appropriate units.

3. Determine the initial speed of the ball, �̇�𝑟. a. Please show the derivation in variable form. That is, your final answer may only be a

function of some or all of the following parameters (𝛼𝛼, 𝑑𝑑, 𝑣𝑣, 𝑔𝑔). b. Please provide the numeric answer based on your equation in part 3a. Please be sure

to include appropriate units.

4. Determine the fixed angle of the tennis ball launcher, 𝜃𝜃. a. Please show the derivation in variable form. That is, your final answer may only be a

function of some or all of the following parameters (𝛼𝛼, 𝑑𝑑, 𝑣𝑣, 𝑔𝑔). b. Please provide the numeric answer based on your equation in part 4a. Please be sure

to include appropriate units.

5. Determine the initial height of the tennis ball launcher, 𝑦𝑦0. a. Please show the derivation in variable form. That is, your final answer may only be a

function of some or all of the following parameters (𝛼𝛼, 𝑑𝑑, 𝑣𝑣, 𝑔𝑔). b. Please provide the numeric answer based on your equation in part 5a. Please be sure

to include appropriate units.

Note that you do not need to include every step in your derivations, but there should be enough so that a reader can easily follow. Also, you should make sure your final equations are in the simplest mathematical form possible. Please note that I have posted Figure 1 on D2L so that you may use it in your project and modify it any way you see fit.