Vectors
The thrust of an airplane's engine produces a speed of 400 mph in still air. The wind velocity is given by
<-20,30>. In what direction should the plane travel to fly due south? Give your answer as an angle from due south.
The airplane velocity should be
(20,-a) where a=sqrt(400^2-20^2)
To find the direction the plane should travel, we need to consider the combined effect of the thrust of the engine and the wind velocity. We can add these vectors to find the resulting velocity.
The thrust vector is in the direction of the plane's travel and has a magnitude of 400 mph. Let's call this vector T.
T = <-400, 0>
The wind velocity vector is given as <-20, 30>. This means that the wind is blowing in the southwest direction with a magnitude of 40 mph.
Now, to determine the resultant velocity, we add the thrust vector and the wind velocity vector:
Resultant velocity = T + Wind velocity
R = <-400, 0> + <-20, 30>
R = <-420, 30>
To find the direction, we can calculate the angle that the resultant velocity makes with due south. We can use the arctan function to find this angle.
Angle = arctan(Ry / Rx)
Angle = arctan(30 / -420)
Angle ≈ -4.3 degrees
Therefore, the plane should travel at an angle of approximately 4.3 degrees west of due south to counteract the effect of the wind and fly due south.