A pilot heading his polane due north finds the actual direction of travel to be 5.2 NE. The planes's air speed is 315 mi/h, and the wind is from die west. Find the plane's ground speed and the velocity of the wind.
Set up a triangle.
If the ground speed is v, then that is the northward component of the velocity. The wind speed w is the eastbound component. Add them to get the air speed. Your triangle shows that
v/315 = cos 5.2°
w/315 = sin 5.2°
Plug and chug
To find the plane's ground speed and the velocity of the wind, we need to break down the given information and solve for these two variables. Here's how we can approach the problem:
1. Start by visualizing the situation. Imagine a coordinate system with north as the positive y-axis and east as the positive x-axis. Draw a vector representing the actual direction of travel (acting as the resultant vector) from the plane's starting point.
2. The pilot's desired direction of travel is due north, but due to the wind blowing from the west, the plane is heading 5.2° northeast of north. Draw the pilot's intended direction of travel, and notice that the resultant vector is slightly bent due to the influence of the wind.
3. Break down the vectors. We have two vectors here: the plane's airspeed vector (acting due north) and the wind vector (from the west). We want to find the ground speed (magnitude of the resultant vector) and the wind's velocity (angle and direction).
4. Apply vector addition. The resultant vector is the sum of the airspeed vector and the wind vector. To find the ground speed, we need to find the magnitude of the resultant vector. To find the wind's velocity, we need to find the angle between the resultant vector and the north direction.
5. Use trigonometry to solve for the magnitude of the resultant vector. Since we have the magnitudes of the airspeed vector (315 mi/h) and the angle between them (5.2°), we can use trigonometric functions (sine and cosine) to find the magnitude of the resultant vector.
6. Use inverse trigonometric functions to solve for the angle between the resultant vector and the north direction. Since we know the angle between the resultant vector and the north direction is 5.2°, we can subtract this angle from 90° to find the angle between the resultant vector and the standard north axis.
7. Calculate the ground speed and the wind's velocity. The ground speed is the magnitude of the resultant vector, which we found in step 5. The wind's velocity is the angle between the resultant vector and the standard north axis, which we found in step 6.
By following these steps, you should be able to find the plane's ground speed and the velocity of the wind.