Physics 109n Homework #1

Due 2:00p.m., Tuesday 10 September.

Thales' tricks:

2500 years ago, Thales brought back to Greece from Egypt some practical ways to use geometrical ideas. In particular, he showed how to find the height of a pyramid by measuring its shadow, and how to find the distance of a boat at sea by measuring line of sight angles from two places on the shore a known distance apart. To see exactly what he did, it's simplest just to do it.

These simple ideas are closely related to astronomical methods, for example finding the distance to the moon and the sun.

1. Find the height of a lamp post in the parking lot behind the Physics Building by measuring the length of its shadow, and measuring the length of your own shadow. Explain clearly how knowing your own height means you can figure the height of the lamp post.

Read both questions 2 &3 before starting to do 2

2. Since we don't have a handy boat at sea, we settle for measuring the distance of the Rotunda from the next path down across the Lawn (not the path right by the Rotunda). Go to one end of that next path down, and measure the angle between the line of the path and a pencil, say, pointing straight to the middle of the Rotunda. Now do the same from the other end of the path. Next, pace off the path, after measuring the length of your pace. Now draw a triangle with baseline to represent the path, in some units you decide, such as 1 inch = 100 feet, (state what unit you decide to use!) draw the other two sides to represent lines from the ends of the path to the middle of the Rotunda. Use a protractor to get the angle equal to what you measured at the Lawn, then use your drawing to measure the distance of the Rotunda from the path.

3. We can now measure the height of the Rotunda without getting close to it. The triangle you drew for question 2 above was to be used to find the distance of the Rotunda from the midpoint of the path. For this question, you must actually go to the midpoint of the path and there measure the angle between a pencil pointing at the topmost point of the Rotunda and the horizontal. Now, back to the drawing board, draw a triangle having as baseline the line from the midpoint of the path to the middle of the Rotunda, and having the angle you just measured (so the other point in the triangle is the top of the Rotunda). If you drew this triangle accurately, the short side of the triangle is the height of the Rotunda -- but, in writing this up, you should make clear why this is so.

An irrational argument

4. To test your understanding of the Pythagoreans' argument that the square root of 2 is not a ratio of two whole numbers, see if you can prove the square root of 3 isn't either, that is, there are no two whole numbers m, n such that m2 = 3n2. HINT: the argument is a lot like that for 2, but this time is not in terms of odd numbers and even numbers. Can you see what the corresponding sets of numbers are this time? What does m2 = 3n2 tell us about possible factors of m?