An airplane undergoes the following displacements: First, it flies 40 km in a direction 30° east of north. Next, it flies 56 km due south. Finally, it flies 100 km 30° north of west. Using analytical methods, determine how far the airplane ends up from its starting point.

The attached Figure shows the plane trajectories from start point (0,0) to (x1,y1) (d1=40km), then going from (x1,y1) to (x2,y2) (d2=56km), then from (x2,y2) to (x3,y3) (d3=100). Taking into account the angles and triangles formed (shown in the Figure), it can be said:

[tex]x1=d1*cos(60), y1=d1*sin(60)\\\\ x2=x1 , y2=y1-d2\\\\ x3=x2-d3*cos(30) , y3=y2+d3*sin(30)[/tex]

Using the Pitagoras theorem, the distance from (x3,y3) to the start point can be calculated as:

[tex]d=\sqrt{x3^{2} +y3^{2} }[/tex]

Replacing the given values in the equations, the distance is calculated.