Homework 9. Finding Distance With Parallax

When your point of view changes, nearby objects appear to shift with respect to more distant ones. This is called parallax, and it's a basic tool for measuring astronomical distances. The same idea can be used to measure distances to objects on Earth.

For this assignment, you need to measure small angles with an accuracy of about 0.1°. An ordinary protractor is not that accurate, but you can easily make something which is. You need a plastic ruler marked in centimeters, a length of thread, and a paperclip. Tie one end of the thread to the paperclip. Starting at the knot, measure out 57 cm of thread, and mark the thread at that point. Place the ruler face-down and lay the thread across the middle, with the mark at the bottom edge of the ruler. Put a piece of tape on the back of the ruler to hold the thread in place. Done!

To use this contraption, take the ruler in one hand and the paperclip in the other. Hold the paperclip to your cheek below your eye, and extend your other arm until the thread is taut. Look toward the objects you want to measure and hold the ruler at a right angle to your line of sight. With the ruler 57 cm from your eye, an angle of 1° corresponds to a separation of 1 cm. Thus, when you sight past the ruler, two objects which appear to be separated by 1 cm have an angular separation of 1°.

This diagram is an overhead view showing the geometry of a parallax measurement. Such a measurement requires observations from two different places separated by a known distance. This distance, the baseline, is represented by the symbol b. Pick a fairly nearby target which you can view in front of a background much further away (for example, you might use the pole of a streetlight as your target, with the side of the valley as a background). For the first observation, line the target up with some definite landmark in the background (for example, a rock on the side of the valley). Now move to your second observation point, and use the ruler-and-thread to measure the angle between your target and the background landmark. The distance R to your target is

The key to this assignment is not just to make a measurement -- you will also have to make some choices, and explain why you made those choices. At every step, your choices affect the accuracy of your result, so think carefully before choosing.

1. Describe the target you chose and the background landmark you will use for your measurement.

2. Pick a value for the baseline distance b, and explain your choice. Ease of measurement is a valid reason for picking a certain baseline, but accuracy is also a consideration. If your baseline is too small, it will be hard to measure the angle accurately. (Note that it does not matter what units you use for b; you will automatically get R in the same units!)

3. Now record the angle you measured, and use the equation on the other side of the page to find R.

4. Finally, try to guess the size of any errors you might have made in measuring b and . Which part of the assignment was most difficult? How would you improve your measurements?