The captain of a plane wishes to proceed due west. The cruising speed of the plane is 255 m/s relative to the air. A weather report indicates that a 39.0-m/s wind is blowing from the south to the north. In what direction, measured with respect to due west, should the pilot head the plane?
He should head S of W by an angle of
arctan39/255
Or she should call off the trip until the weather is calmer.
To determine the direction in which the pilot should head the plane, we need to find the resulting velocity that combines the plane's velocity and the wind's velocity.
Step 1: Visualize the problem
Draw a diagram to represent the situation. Draw a horizontal line to represent due west (let's call it x-axis), and a vertical line (let's call it y-axis) to represent the north-south direction.
Step 2: Assign variables
Let's assign variables to the known values:
- Plane's speed relative to air = 255 m/s
- Wind speed = 39.0 m/s
- Angle of plane's heading with respect to due west = θ (what we need to find)
Step 3: Break down the velocities
Since the wind is blowing from south to north, it affects the plane's velocity. We need to break down both the plane's velocity and wind's velocity into x and y components.
The x-component represents motion in the east-west direction (parallel to the x-axis),
and the y-component represents motion in the north-south direction (parallel to the y-axis).
The plane's x-component velocity (Vp-x) is its own velocity relative to air, which is 255 m/s (due west).
The plane's y-component velocity (Vp-y) due to wind blowing north is 0 m/s (no effect on north-south motion).
The wind's x-component velocity (Vw-x) is 0 m/s (no effect on east-west motion).
The wind's y-component velocity (Vw-y) is its own velocity, which is 39.0 m/s (northward).
Step 4: Combine the velocities
The resultant x-component velocity (Vr-x) is the sum of the plane's and wind's x-component velocities:
Vr-x = Vp-x + Vw-x = 255 m/s + 0 m/s = 255 m/s
The resultant y-component velocity (Vr-y) is the sum of the plane's and wind's y-component velocities:
Vr-y = Vp-y + Vw-y = 0 m/s + 39.0 m/s = 39.0 m/s
Step 5: Find the angle
To find the angle θ (measured with respect to due west), we can use the tangent function:
tan(θ) = Vr-y / Vr-x
Rearranging the equation, we get:
θ = atan(Vr-y / Vr-x)
Substituting the values:
θ = atan(39.0 m/s / 255 m/s)
Using a calculator, the value of θ is approximately 8.4 degrees.
The pilot should head the plane approximately 8.4 degrees north of due west to compensate for the wind.