A flock of ducks is trying to migrate south for the winter, but they keep being blown off course by a wind blowing from the west at 5.0 m/s . A wise elder duck finally realizes that the solution is to fly at an angle to the wind. If the ducks can fly at 7.0 m/s relative to the air, what direction should they head in order to move directly south?

The ducks' flight path as observed by someone standing on the ground is the sum of the wind velocity and the ducks' velocity relative to the wind:

ducks (relative to wind) + wind (relative to Earth) = ducks (relative to Earth)

or equivalently,

(see the attached graphic)

We have

Then by setting components equal, we have

We only care about the direction for this question, which we get from the first equation:

or approximately 136º or 224º.

Only one of these directions must be correct. Choosing between them is a matter of picking the one that satisfies both equations. We want

which means θ must be between 180º and 360º (since angles in this range have negative sine).

So the ducks must fly (relative to the air) in a direction 224º relative to the positive horizontal direction, or about 44º South of West.

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