What heading (or direction) and airspeed (speed in still air) are required for an airplane to fly 837 mph due north if a wind of 63 mph is blowing in the direction of S 11.5 degrees E?
To determine the heading and airspeed required for an airplane to fly at a specific ground speed and direction, we can use the concept of vectors and vector addition.
1. First, let's consider the effect of the wind on the airplane's ground speed and direction. The wind is blowing at a speed of 63 mph in the direction of S 11.5 degrees E.
2. To break down the wind vector into its east-west (crosswind) and north-south (headwind/tailwind) components, we can use trigonometry. The component of the wind blowing in the north-south direction can be found by multiplying the wind speed (63 mph) by the sine of the wind direction (11.5 degrees):
North-South Component = 63 mph * sin(11.5 degrees)
3. Similarly, the east-west component of the wind can be found by multiplying the wind speed (63 mph) by the cosine of the wind direction (11.5 degrees):
East-West Component = 63 mph * cos(11.5 degrees)
4. Now, let's analyze the airplane's motion relative to the wind to determine the required heading and airspeed.
a) To maintain a ground speed of 837 mph due north, the airplane needs to overcome the headwind component of the wind and have an airspeed equal to the ground speed plus the headwind component. This will cancel out the effect of the headwind and maintain the desired ground speed.
Airspeed = Ground Speed + North-South Component of Wind
b) The heading (or direction) of the airplane will be such that, when combined with the crosswind component of the wind, it cancels out the effect of the crosswind. This means that the resultant vector of the airplane's velocity and the wind vector should point directly north.
Heading = arctan(East-West Component of Wind / (Ground Speed + North-South Component of Wind))
5. Calculate the values using the given information:
North-South Component = 63 mph * sin(11.5 degrees) ≈ 13.32 mph
East-West Component = 63 mph * cos(11.5 degrees) ≈ 61.48 mph
Airspeed = 837 mph + 13.32 mph ≈ 850.32 mph
Heading = arctan(61.48 mph / 850.32 mph) ≈ 4.09 degrees
Therefore, to fly 837 mph due north with a wind of 63 mph blowing in the direction S 11.5 degrees E, the airplane would require an airspeed of approximately 850.32 mph and a heading of approximately 4.09 degrees.