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 6.0m/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 8.0m/s relative to the air, what direction should they head in order to move directly south? (answer is in degrees west of south)
Kill the birds
Jaymz! How dare you, I thought we raised you better than this...
You get a bird, you get a bird, EVERYONE GETS A BIRD
The have to have a 6.0m westward component to counter the wind, (draw the triangle) so the S component of their velocity is sqrt (8^2-6^2)=sqrt (64-36)
and the direction is.... arcsin 6/8 west of S.
To determine the direction the ducks should head, we need to consider the vector addition of their velocity relative to the air and the wind velocity. Here's how you can calculate it:
1. Assign positive values to the east and north directions. Since we want the direction relative to the south, let's assign the south direction a negative value.
2. Convert the wind velocity from west to east by considering it as a negative value since it blows towards the east.
3. Identify the components of the velocity vectors. The ducks' velocity relative to the air is 8.0 m/s south (+8.0 m/s), and the wind velocity is 6.0 m/s west (-6.0 m/s).
4. Add the components in the east-west (x-axis) and north-south (y-axis) directions.
- The east-west component is -6.0 m/s (from the wind) since it is blowing towards the east.
- The north-south component is +8.0 m/s (from the ducks' velocity relative to the air) since they are flying south.
- Add the east-west and north-south components together: -6.0 m/s + 8.0 m/s = 2.0 m/s.
5. Use the resultant east-west and north-south components to calculate the direction using trigonometry.
- The angle (θ) can be calculated using the inverse tangent function: θ = atan(east/west).
- In this case, θ = atan(2.0 m/s / 8.0 m/s).
- Calculate θ using your calculator or a trigonometric table to get the result.
- The direction relative to the south is given by the angle west of south (180° - θ).