The problem asks:

In still air, a bird can fly at a speed of 10 m/s. The wind is blowing from the east at 6 m/s.
In which direction must the bird fly in order to travel to a destination that is due north of the bird's current location?

Options: A. $37^\circ$ east of north
B. $37^\circ$ west of north
C. $53^\circ$ east of north
D. $53^\circ$ west of north

Solution:

This is a classic vector problem involving relative velocity. The bird's velocity relative to the ground needs to be directly north, so we need to resolve the velocity of the bird against the wind.

1. Wind velocity is 6 m/s toward the west (because wind blows from the east).
2. Bird's airspeed is 10 m/s in some direction (which we need to determine).
3. The resultant velocity of the bird relative to the ground must be directly north (0 m/s east-west component).

Let’s resolve the components:

• Let the bird fly at an angle $\theta$ west of north (since the wind is blowing from the east). The bird's velocity has two components:
  • $V_{\text{bird, north}} = 10 \cos(\theta)$ (northward component)
  • $V_{\text{bird, east-west}} = 10 \sin(\theta)$ (east-west component)

Since the wind velocity is 6 m/s toward the west, we set the sum of the east-west components to zero for the bird to fly directly north:

$10 \sin(\theta) = 6$

Solving for $\sin(\theta)$:

$\sin(\theta) = \frac{6}{10} = 0.6$

Now, using the inverse sine function:

$\theta = \sin^{-1}(0.6) \approx 37^\circ$

So, the bird should fly $37^\circ$ west of north to counter the wind and fly directly north.

Thus, the correct answer is B. $37^\circ$ west of north.

Do you want further details or have any questions?

Here are some related questions for practice:

1. How do you calculate the relative velocity in two dimensions?
2. If the wind speed increased, how would the angle change for the bird to maintain a northward path?
3. What would happen if the bird flew slower than 6 m/s?
4. How can this problem be solved using graphical vector addition?
5. Can this method apply to other vehicles like airplanes or ships?

Tip: Always break down velocity problems into their component vectors for easier calculation.