A pilot is flying a Cessna airplane at 180 mph (airspeed).He would like to fly in the direction N45W, but there is a 32 mph wind in the S60W direction.What direction should the pilot set his course for in order to fly along his desired track? What will his speed relative to the ground be?

To find the direction the pilot should set his course and the speed relative to the ground, we can use vector addition.

Step 1: Convert the given wind speed and direction into components.
The wind is blowing at 32 mph in the S60W direction. To convert this into components, we need to break it down into its north/south (S) and east/west (W) components.
The S60W direction can be represented as a vector with components (-sin(60), -cos(60)), where the negative sign indicates the wind is blowing in the opposite direction. We can simplify this to (-√3/2, -1/2).
Multiplying these components by the magnitude of the wind speed (-32 mph), we get the wind vector (-16√3, -16).

Step 2: Calculate the course correction.
Since the wind is opposing the desired direction of N45W, the pilot needs to adjust his course to compensate for the wind's effect. We can treat the pilot's desired direction as a vector as well.
The N45W direction can be represented as a vector with components (-sin(45), sin(45)), since N is positive for the northward component and W is negative for the westward component. Simplifying this, we get the vector (-√2/2, -√2/2).
Adding the wind vector and the desired direction vector together, we get the overall vector. (-16√3 - √2/2, -16 - √2/2)
Simplifying, the overall vector is (-√2/2 - 16√3, -√2/2 - 16).

Step 3: Find the direction of the overall vector.
To determine the direction of the overall vector, we can calculate its angle with respect to the north direction using trigonometry.
The angle θ can be found using the equation tan(θ) = (southward component)/(northward component).
tan(θ) = (-√2/2 - 16) / (-√2/2 - 16√3)
Simplifying, θ ≈ -42.5°.

Therefore, the pilot should set his course to N42.5W in order to fly along his desired track.

Step 4: Calculate the speed relative to the ground.
To find the speed relative to the ground, we need to calculate the magnitude of the overall vector.
The magnitude of the overall vector can be calculated using the Pythagorean theorem.
Magnitude = sqrt((northward component)^2 + (westward component)^2)
Magnitude = sqrt(((-√2/2 - 16√3))^2 + ((-√2/2 - 16)^2))
Magnitude ≈ 161.5 mph.

Hence, the pilot's speed relative to the ground would be approximately 161.5 mph.

Draw a diagram of the velocities. If the plane's heading is θ, we need the final x- and y-components to be N45W. So,

180sinθ - 32cos60 = v sin135
180cosθ - 32sin60 = v cos135

θ = 20°
v^2 = 180^2+32^2-2*180*32 cos40°
v = 157 mph