# Group Project II

Guidelines: You should consider your project as a technical paper (like an English paper only different). Part of your grade will be based on the quality of your written presentation. The paper you turn in should have a mix of equations, formulas and prose. Graphs may be copied from your calculator, but should be clearly labelled. Use complete sentences, good grammar and correct punctuation. Spelling is also important. You should aim to write your report in such a way that it can be read and understood by anyone who knows the material for this course. Finally, as a general rule for this and any other project, whenever you hypothesize an answer to a question, be sure to provide ample justification as to why you believe your answer to be correct.

You may have read in your newspaper a few weeks ago that Spaceman Spiff and his friend Hobbes are planning to pull Mars out of its orbit and put it on a collision course with Earth. But before setting out on their mission, Spiff needs some critical information about the motions of Earth and Mars relative to each other. He has hired your group to provide that information. He's paying you big money, so be sure to include arguments to convince him that your answers are correct.

Hobbes informs you that the Earth is 93 million miles from the sun, while Mars is 141 million miles from the sun. The two planets orbit the sun in the same direction, with Mars taking 1.88 years to complete one orbit. (You may assume the orbits of the planets are circular with the sun at the center.) On April 1, 1994 they were 48 million miles apart, which is as close as they can get. And on January 1, 1994 they were 95.3 million miles apart.

Here are Spaceman Spiff's instructions:

### Part I

First, find expressions for the positions of Earth and Mars in terms of time. Let D(t) be the distance between the two planets at time t.
• Find a formula for D(t). Support you answer by showing that your formula passes as many tests as you can think of.
• How far apart were the planets on October 1?
• What was the average rate at which the distance between the planets was changing from April 1 to October 1? Express your answer in miles per hour. How about from September 1 to October 1? From September 30 to October 1?
• At what speed were the planets moving away from or toward each other on October 1? Were they getting closer together or farther apart?

Spiff and Hobbes are no dummies. Their plan is to crash the two planets together when they are already moving toward each other at the maximum possible speed. Spiff needs to know when this will next occur. He also needs to know at what speed (in mph) they will be moving toward each other at this time.

(Hints: To find appropriate expressions for the positions of Mars and Earth, you may want to refer to pages 66-68 of your text. You may also want to make use of the distance formula: the distance between two points (a,b) and (c,d) is {(a-c)^2+(b-d)^2}1/2

### Part II

Give a physical explanation why D(t) must be a periodic function. What is the period of D(t)? Find a function of the form A.sin(Bt+C)+E which you think best fits the graph of D(t). As a way of seeing how well your function fits D(t), graph the difference between the two functions.

During one period of D(t), for how long is D(t) within five million miles of its maximum value? Compare this with how long it is within 5 million miles of its minimum value. Based on your answers, do you think it's possible to find a function of the form A.sin(Bt+C)+E which fits D(t) exactly? Why or why not?

### Part III

Let H(t) be the function of the form A.sin(Bt+C)+E you found in Part II. You may be wondering what Spiff has in mind for this function. Well, nothing really, except that he wants you to draw a rough sketch of the derivative of this function over one period using primarily visual techniques and perhaps a few calculations. Can you find a formula for this graph? How does the amplitude of this graph compare with the amplitude of H(t)? Tell Spiff everything interesting you can think of about this function.