The captain of a plane wishes to proceed due west. The cruising speed of the plane is 226 m/s relative to the air. A weather report indicates that a 32.0-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?

Question

The captain of a plane wishes to proceed due west. The cruising speed of the plane is 226 m/s relative to the air. A weather report indicates that a 32.0-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?

Answer 1

Vp + Vw = -226 m/s.

Vp + 32i = -226
Vp = -226 - 32i

Tan Ar = -32/-226 = 0.14159
Ar = 8.06o = Reference angle.
A = 8.06 + 180 = 188.06o,Q3. = 8.06o S. of W. = Direction.

Answer 2

To find the direction in which the captain should head the plane, we need to consider the effect of the wind on the plane's velocity.

Let's break down the problem into components. We have the plane's velocity relative to the air, which is due west, and the wind blowing from the south to the north. We can resolve the velocities into their east-west (x) and north-south (y) components.

Given:
Cruising speed of the plane relative to the air (v_air) = 226 m/s
Velocity of the wind blowing from south to north (v_wind) = 32.0 m/s

Since we are looking for the direction of the plane, we need to find the resulting velocity of the plane, which is the vector sum of the plane's velocity relative to the air and the wind's velocity.

Let's define the positive x-direction as due west and the positive y-direction as due south (opposite to the wind direction).

To find the x-component of the resulting velocity (v_x_resultant), we subtract the wind velocity from the plane's velocity relative to the air:
v_x_resultant = v_air - v_wind

v_x_resultant = 226 m/s - 32.0 m/s
v_x_resultant = 194.0 m/s

The resulting x-component of the velocity is 194.0 m/s towards the west.

To find the y-component of the resulting velocity (v_y_resultant), we add the wind velocity to the plane's velocity relative to the air:
v_y_resultant = 0 + v_wind

v_y_resultant = 0 m/s + 32.0 m/s
v_y_resultant = 32.0 m/s

The resulting y-component of the velocity is 32.0 m/s towards the south.

Now, we can find the direction of the resulting velocity vector (θ_resultant) using the arctan function:

θ_resultant = arctan(v_y_resultant/v_x_resultant)

θ_resultant = arctan(32.0 m/s / 194.0 m/s)

Using a calculator or trigonometric tables, we find that θ_resultant ≈ 9.44 degrees.

Therefore, the pilot should head the plane in a direction approximately 9.44 degrees north of due west to counteract the effect of the wind and maintain a true due west heading.