# Math 224--Differential Equations: Project 1

## ...A matter of law

### by Gavin LaRose (glarose@umich.edu), Nebraska Wesleyan University, January 1996

permission granted to use and distribute free in an academic setting

As recently hired members of a successful independent mathematical and scientific technical contracting company, you have just received notice of your first contract. The letter you received from Hangemhi, Inc., a legal consulting company known to have some affiliation with the world-wide MPC, Inc. conglomerate, follows.

plain TeX file with project -- this uses a blackletter font
PostScript file of project

## The letter...

### Hangemhi, Inc.

Lonlinc, SK
04685

26 January 1996

Rigorous Mathematical Contractors (RiMaC), Inc.
Lonlinc, SK 04685

Dear RiMaC:

In the course of our latest legal entanglement, we are in the position of defending a client who is purported to have lept (for reasons which, you will clearly understand, we are not in a position to speculate) from a window of a building. So as to obtain all possible insight into this possibility (however slight), we are attempting to determine the velocity that our client might have attained in the course of allegedly plummeting from one of the windows in question to the ground below. Unfortunately, owing to the busy nature of the street, we are unable to directly measure the velocity a projectile of the appropriate weight (approximately 180 pounds) would attain in the course of the alleged descent---on account, of course, of the possible impingement on the safety of those who might be found in the street below and resulting legal complications.

We therefore need your assistance to obtain a mathematical solution to the problem of what the velocity of our client would be if (s)he were to have fallen from a first story window (approximately 5 feet from the ground), second story window (approximately 15 feet from the ground), a third story window, and, for good measure, a fifth story window as well. In that one of our partners is an accomplished sky-diver, we are certain from her experience that the velocity will not exceed the terminal (no pun intended) velocity of 120 miles per hour, but suspect that the velocity of our client will be rather lower than that.

Your physical measurements department has determined for us that the force on a falling body (or, as the case might be, other projectile) may be modeled by f = mg - Kv^2, where g is the acceleration due to gravity, m the mass of the body, and v its velocity, but did not provide a solution to the problem as stated.

As specified in your contract, your report should be submitted by the 16th of next month, as the case is to go to court the following week. If you should find in the course of your investigation that you have questions regarding this project, you are to contact Dr. Gavin LaRose, our firm's consulting scientist (who is, unfortunately, only working for us part-time around his job with Chemproc, Inc., and hence is unable to resolve this problem for us directly) with the other members of your investigative team; he will be more than willing to assist you. You should further contact him (also as a team) by the 9th of February to report on your progress---failure to do so will result in a substantial payment penalty.

We look forward to seeing your finished report, which should conform to the enclosed technical report requirements.

Yours sincerely,
Claire N. ``CD'' Arro, Partner
Hangemhi, Inc.

Encl: technical report requirements

## The technical report requirements...

### Hangemhi, Inc.

Technical Report Requirements

It is of paramount importance that the reports submitted to Hangemhi, Inc. be written so that all members of the legal team using the report can understand and apply the information contained therein. These legal team members are all experts in law, but have little recent experience in mathematics. They should therefore be assumed to remember basic differential and integral calculus, but little more.

Reports should further:

• Be written in the first person plural.
• Include mathematical formulas in the report as appropriate to describe the methods used and results obtained.
• Clearly explain how the mathematical formulas that are included bear on the problem being solved, and how the solution(s) obtained answer(s) the question(s) that the report is to answer.Clearly explain how the mathematical formulas that are included bear on the problem being solved, and how the solution(s) obtained answer(s) the question(s) that the report is to answer.
• Consist of
• An Introduction, indicating generally the problem(s) to be solved, and giving an explanation of the organization of the remainder of the report.
• A section for each of the problems considered, each having
• An Introduction, specifically describing the problem to be solved, and an indication of the mathematical method used to solve it.
• A Body, describing the mathematical problem that was solved to answer the question(s) posed in the introduction, and the solution to it.
• A Conclusion, summarizing the results obtained from the solution described in the body and clearly stating their relevance to the original problem as described in the introduction.
• Be 2.5--5 pages in length.

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