A pilot wants to maintain a course of 40° and a ground speed of 300 mph against a 45 mph wind from 20° west of north. What should his heading be?

To calculate the pilot's heading, we need to consider the wind's effect on the aircraft's course and ground speed.

1. Calculate the wind vector:
- The wind is blowing from 20° west of north, which means it has an angle of 90° + 20° = 110° from the north.
- The wind speed is given as 45 mph.
- Convert the wind speed and angle to vector components using trigonometry:
- The north component of the wind vector is 45 * cos(110°).
- The east component of the wind vector is 45 * sin(110°).

2. Calculate the aircraft's heading:
- The desired course is 40°.
- The ground speed is given as 300 mph.
- To maintain the desired course, the aircraft needs to counteract the wind by pointing its nose slightly to the left of the desired course.
- The angle between the heading and the desired course is called the crab angle or drift angle.
- The heading is the sum of the desired course and the drift angle.

3. Calculate the drift angle:
- To calculate the drift angle, we use the wind vector components:
- The drift angle (in radians) is the arc tangent (atan) of the east component divided by the ground speed.
- Convert the drift angle from radians to degrees using the conversion factor (180° / π).
- Subtract this drift angle from the desired course to obtain the heading.

Let's calculate the values step by step:

1. Calculate the wind vector:
- The north component of the wind vector is 45 * cos(110°) = -28.28 mph (-cosine is negative in the southern hemisphere).
- The east component of the wind vector is 45 * sin(110°) = -38.96 mph (-sine is negative in the third quadrant).

2. Calculate the aircraft's heading:
- The drift angle = atan(-38.96 / 300) * (180° / π) = -7.86°.
- The heading = desired course + drift angle = 40° - 7.86° = 32.14°.

Therefore, the pilot should maintain a heading of approximately 32.14° to maintain a course of 40° and a ground speed of 300 mph against a 45 mph wind from 20° west of north.

To determine the pilot's heading, we need to account for the wind direction and speed.

Step 1: Calculate the wind components:
Given that the wind is coming from 20° west of north, the angle between the wind direction and the north is 90° - 20° = 70°.
Using trigonometry, we can find the two components of the wind:
- Crosswind component (perpendicular to the desired course): wind speed * sin(angle between wind direction and course)
- Headwind component (opposite to the desired course): wind speed * cos(angle between wind direction and course)

In this case:
Crosswind component = 45 mph * sin(70°)
Headwind component = 45 mph * cos(70°)

Step 2: Find the effective wind speed and direction:
The effective wind speed and direction are the components that directly affect the aircraft's motion.
To find the effective wind speed, we'll use the Pythagorean theorem:
Effective wind speed = square root of (Crosswind component^2 + Headwind component^2)

The effective wind direction is the direction opposite to the resultant wind vector. To find it, we'll use the inverse tangent function:
Effective wind direction = arctan(Crosswind component / Headwind component)

Step 3: Calculate the pilot's heading:
The pilot's heading will be the sum of his desired course and the effective wind direction:
Heading = Desired course + Effective wind direction

Let's calculate the values:

Crosswind component = 45 mph * sin(70°) = 41.39 mph
Headwind component = 45 mph * cos(70°) = 17.66 mph

Effective wind speed = square root of (41.39^2 + 17.66^2) = 45.66 mph (approximately)
Effective wind direction = arctan(41.39 / 17.66) = 65.45° (approximately)

Heading = 40° + 65.45° = 105.45°

Therefore, the pilot's heading should be approximately 105.45°.

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