Suppose that a wind is blowing in the direction S45°E at a speed of 30 km/h. A pilot is steering a plane in the direction N60°E at an airspeed (speed in still air) of 100 km/h. The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and the wind. The ground speed of the plane is the magnitude of the resultant. Find the true course and the ground speed of the plane.

[tex]V_w[/tex] = velocity of wind = [tex]30\ km/h\ S45^\circ E = (30\cos 45^\circ\ \hat{i}-30\sin 45^\circ\ \hat{j})\ km/h = (21.21\ \hat{i}-21.21\ \hat{j})\ km/h[/tex]

[tex]V_p[/tex] = velocity of plane in still air = [tex]100\ km/h\ N60^\circ E = (100\cos 60^\circ\ \hat{i}+100\sin 60^\circ\ \hat{j})\ km/h = (50\ \hat{i}+86.60\ \hat{j})\ km/h[/tex]

Assume:

[tex]V_r[/tex] = resultant velocity of the plane

[tex]\theta[/tex] = direction of the plane with the east

Since the resultant is the vector addition of all the vectors. So, the resultant velocity of the plane will be the vector sum of the wind velocity and the plane velocity in still air.

[tex]\therefore V_r = V_p+V_w\\\Rightarrow V_r = (50\ \hat{i}+86.60\ \hat{j})\ km/h+(21.21\ \hat{i}-21.21\ \hat{j})\ km/h\\\Rightarrow V_r = (71.21\ \hat{i}+65.39\ \hat{j})\ km/h[/tex]

Let us find the direction of this resultant velocity with respect to east direction:

[tex]\theta = \tan^{-1}(\dfrac{65.39}{71.21})\\\Rightarrow \theta = 40.29^\circ[/tex]

This means the the true course of the plane is in the direction of [tex]40.29^\circ[/tex] north of east.

The ground speed will be the magnitude of the resultant velocity of the plane.

[tex]\therefore Magnitude = \sqrt{71.21^2+65.39^2} = 96.68\ km/h[/tex]

Hence, the ground speed of the plane is 96.68 km/h.