A pilot checks her instruments and finds that her airspeed is 345 km/h and that she is heading in a direction [S40ºW].  A radio report tells her that the wind velocity is 85 km/h [NW].  What is the velocity of her plane relative to the ground?

I tried to draw this out, but I think i did it wrong. I also tried to break it into vector components, but I am not understanding

the plane is moving inside an air mass that is moving relative to the ground

the ground speed is the sum (vector) of the two speeds

345 kph, S40ºW
... 345 cos(40º) S
... 345 sin(40º) W

85 kph, NW
... 85 cos(45º) N
... 85 sin(45º) W

find the vector sum of the 4 components

To solve this problem, we need to break down the velocities into their respective components and then add or subtract them to find the resultant velocity.

Let's start by breaking down the given velocities into their components:

1. Airspeed: 345 km/h at a heading of [S40ºW]
- The given heading tells us that the plane is heading 40 degrees west of the south direction.
- To find the south and west components of the velocity, we need to use trigonometry.
- The southward component of the airspeed will be 345 km/h multiplied by the cosine of the angle (40 degrees).
- The westward component will be 345 km/h multiplied by the sine of the angle (40 degrees).

So, the southward component is: 345 km/h × cos(40°)
And the westward component is: 345 km/h × sin(40°)

2. Wind velocity: 85 km/h [NW]
- The direction given (Northwest) is diagonally between the north and west directions.
- We can break down the wind velocity into its northward and westward components using trigonometry.
- The northward component will be 85 km/h multiplied by the cosine of the angle (45 degrees).
- The westward component will be 85 km/h multiplied by the sine of the angle (45 degrees).

So, the northward component is: 85 km/h × cos(45°)
And the westward component is: 85 km/h × sin(45°)

Now that we have the components of both the airspeed and the wind velocity, we can add the respective components to find the resultant velocity. Add the southward component of the airspeed to the northward component of the wind velocity and the westward component of the airspeed to the westward component of the wind velocity.

The resultant velocity is the vector sum of these components and represents the velocity of the plane relative to the ground.