an airplane is flying with an airspeed of 145 mph and a heading of 84.5 degrees. q wind of 24.6 omh is blowing at a heading of 68.4 degrees. find the ground speed and course of airplane

One does not normally give heading of a breeze. You specify the direction it comes FROM. However this seems to be specifying the direction it is GOING like it were a current so I will assume that.

north speed = Vn = 145 cos 84.5 + 24.6 cos 68.4

east Speed = Ve = 145 sin 84.5 + 24.6 sin 68.4

Speed = sqrt(Vn^2 + Ve^2)

course made good = C
tan C = Ve/Vn

To find the ground speed and course of the airplane, we'll need to break down the airspeed and wind velocity into their respective components and then add them together.

Step 1: Resolve the Airspeed
The airspeed can be resolved into horizontal and vertical components using trigonometry. The horizontal component, also known as the groundspeed, can be found by multiplying the airspeed by the cosine of the heading angle. The vertical component is obtained by multiplying the airspeed by the sine of the heading angle.

Horizontal component (groundspeed) = airspeed × cosine(heading angle)
Vertical component = airspeed × sine(heading angle)

Given the airspeed of 145 mph and heading angle of 84.5 degrees:
Horizontal component = 145 mph × cosine(84.5°)
Vertical component = 145 mph × sine(84.5°)

Step 2: Resolve the Wind Velocity
Similarly, we can resolve the wind velocity into horizontal and vertical components using trigonometry. Again, the horizontal component is the one that affects the groundspeed, while the vertical component does not.

Horizontal component = wind velocity × cosine(wind angle)
Vertical component = wind velocity × sine(wind angle)

Given the wind velocity of 24.6 mph and wind angle of 68.4 degrees:
Horizontal component = 24.6 mph × cosine(68.4°)
Vertical component = 24.6 mph × sine(68.4°)

Step 3: Calculate the Groundspeed
To find the groundspeed, we need to add the horizontal components of the airspeed and wind velocity. The formula is:

Groundspeed = Airspeed horizontal component + Wind velocity horizontal component

Given the horizontal components from steps 1 and 2, we have:
Groundspeed = Airspeed horizontal component + Wind velocity horizontal component

Step 4: Calculate the Course
The course is the heading or direction in which the airplane is actually moving. To find it, we need to consider both the heading angle and the wind angle.

Given the heading angle of 84.5 degrees and the wind angle of 68.4 degrees, the course can be found by adding the heading angle to the angle between the groundspeed and the x-axis. This angle can be determined using the inverse tangent function.

Course = (heading angle + arctan(Vertical component / Groundspeed horizontal component)) mod 360°

Now that we have all the necessary information, we can calculate the groundspeed and course of the airplane.