Sail Away!

You are headed for a picnic on Palm Island, which is 5 miles to the east. The wind, unfortunately, is blowing to the south, at 5 miles per hour.

Not to worry! You and your shipmates are master-sailors, and experts at vectors, too!

You point your boat to the east, while angling your sail at a 45° angle to the south-west:

1. If we consider the wind as a vector whose length corresponds to the wind speed in miles per hour, what vector would represent it?

W = ____________

2. If the sail is represented by a unit-vector emanating from the boat, denoted by S, then
S = ____________

3. Now re-draw Figure 1, adding in two vectors, E (the Effective Wind) and N (the Negligible Wind) into which W decomposes. Explain (in words) why only E will affect our sail.

4. Now identify N as a vector projection:
N is the vector projection of ____________ onto____________

Can you give a convincing explanation (in words, or orally) to your shipmates as to why this is true?

1. Find N

2. What two relations (one algebraic, one geometric) hold between E and N?

3. Find E

4. This E represents the windforce which acts (through your sail) on your boat. Find the norm of E, |E| and interpret its physical significance.

5. Due to the keel, no lateral (sideways) motion is possible for your sailboat; only forwards/backwards motion.

By now decomposing E into components L (lateral) and F (forwards/backwards), find the force propelling your boat eastwards. How long will it take you to reach Palm Island by this method (assuming the wind remains constant?)

6. Same questions, for your trip back from Palm Island.

Extra for Expert Sailors

What happens if you angle your sail at a different angle than 45°?

Can you get better results by doing this? (Your answer will depend, of course, on whether you are in a hurry to get to Palm Island, or want to enjoy a leisurely sailing trip!)

What happens if the wind changes to 5mph North? South-West? At some angle ?

©Copyleft 1995 by Harel Barzilai. "Share and Enjoy." Inspired by Chapter 8 of Why Math? by R.D. Driver.