A pilot flies a plane at a speed of 600 km/h with respect to the surrounding air. A wind is blowing to the northeast at a speed of 100 km/h. In what direction must the plane be aimed so that the resulting direction of the plane is due north?
Graphically this is so easy, and that is exactly how pilots would do it.
Put the wind at 100 NE, then on the end of that, take a vector from that point 600 in length, use a compass to draw an arc until it intercepts N. Then that is the direction.
Want to do it analytically?
You have the wind giving a 70.7km/h push to the East, so the plane must cancel that with a 70.7 W component.
so 70.7=600sinTheta (theta measured from N).
sinTheta= 70.7/600
theta= aresin (above) WEST of N
To find the direction in which the plane must be aimed, we need to find the angle between the plane's velocity vector and the north direction. Here's how we can calculate it:
1. First, let's draw a diagram to visualize the situation. Draw two vectors: one representing the velocity of the plane relative to the air, pointing northeast, and another representing the wind velocity, also pointing northeast.
- The plane's velocity vector has a magnitude of 600 km/h, and it is the vector we need to find the angle for.
- The wind's velocity vector has a magnitude of 100 km/h and points northeast.
2. Since the plane's velocity vector is the resultant of its actual speed and the wind speed, we can determine its magnitude and direction using vector addition. Subtract the wind vector from the plane's velocity vector to get the resultant vector, representing the plane's velocity relative to the ground.
3. To find the direction of the resultant vector, you can use trigonometry. Since the resultant vector points due north, it forms a right triangle with the north direction. Label the angle between the resultant vector and the north direction as θ (theta).
4. We can use the tangent function to find the angle θ. Tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle. In this case, the opposite side is the northward component of the resultant vector, and the adjacent side is the eastward component of the resultant vector. Using the tangent function:
- tan(θ) = opposite/adjacent
- tan(θ) = 600 km/h/100 km/h = 6
5. Now, we need to find the angle θ whose tangent is 6. Use the inverse tangent (arctan) function to find θ.
- θ = arctan(6) ≈ 80.54 degrees
6. Therefore, the plane must be aimed at an angle of approximately 80.54 degrees north of due east to counteract the effect of the wind and fly due north.
Note: The actual formula used for vector addition in step 2 is more complex, involving both magnitude and direction angles. However, since we are only interested in finding the angle in this case, we can simplify the calculation.