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?
The information translated into triangle ABC,
where BC is vertical and magnitude 837
AB is our required vector
and angle C = 168.5°
by cosine law
|AB|^2 = 837^2 + 63^2 - 2(837)(63)cos 168.5°
|AB| = 898.82
by sine law:
Sin B / 63 = sin168.5/898.82
sinB = .013974
B = .8 °
To determine the heading and airspeed required for an airplane to fly 837 mph due north, taking into account the wind speed and direction, you need to use the concept of vector addition.
1. Convert the wind direction from its compass notation (S 11.5 degrees E) to a vector notation. Start by drawing a compass rose and locate the south direction (S). From there, measure an angle of 11.5 degrees in the eastward direction (E). This gives you the vector representing the wind direction.
2. Next, calculate the components of the wind vector. Since the wind is blowing to the southeast, the vertical component can be determined using the sine function: sin(11.5) * 63 mph. Similarly, the horizontal component can be determined using the cosine function: cos(11.5) * 63 mph.
3. Now, set up two equations to solve for the required heading and airspeed of the airplane. Let H represent the heading angle and A represent the airspeed.
- For the north component, A * sin(H) - vertical component of wind = 837 mph
- For the east component, A * cos(H) - horizontal component of wind = 0 mph (since we want to fly due north)
4. Substitute the calculated values for the wind components into the equations:
- A * sin(H) - (sin(11.5) * 63) = 837
- A * cos(H) - (cos(11.5) * 63) = 0
5. You can solve these two simultaneous equations to find the values of the heading (H) and airspeed (A) required for the airplane to fly 837 mph due north. This can be done either manually using algebraic techniques or by using numerical methods like iteration or a software tool like a spreadsheet or equation solver.
By following these steps, you can determine the heading and airspeed needed for the airplane to travel at 837 mph due north, taking into account the given wind speed and direction.