Question

An aircraft pilot wishes to fly from an airfield to a point lying S20oE from the airfield. There is a wind blowing from N80oE at 45 km/h. The airspeed of the plane will be 550 km/h.
(a) What direction should the pilot steer the plane (to whole degree)? Include a diagram as part of your solution.
(b) What will the actual ground speed be of the plane (to one decimal place)?

Answer 1

in x,y coordinates, angles counterclockwise from x
Planes flies 550 at angle T
air speed north (y) = 550 sin T
air speed east (x) = 550 cos T

current flow is 45 km/h 10 degrees south of west
current speed north (y) = -45 sin 10 = - 7.81
current speed east (x) = -45 cos 10 = -44.3

total ground speed North (y) = 550 sin T -7.81
total ground speed East (x) = 550 cos T - 44.3

in the end we need to make angle S 20 E which is 270+20 = 290 in x y
tan 290 = -2.75 which has to be our north ground speed / east ground speed
[550 sin T -7.81] / [550 cos T - 44.3] = -2.75
550 sin T - 7.81 = -1513 cos T + 122
550 sin T + 1513 cos T = 130
550 [sqrt(1-cosT^2)] + 1513 cosT = 130
let z = cos T
550 [sqrt(1-z^2)] + 1513 z = 130
sqrt (1-z^2) =- 2.75 z + .236
1-z^2 = 7.56 z^2 - 1.3 z + .0557
8.56 z^2 -1.3 z - .944 = 0
z = 0.417 or -0.265 = cos T
cos is + in quad 1 and 4, - in quad 2 and 3
We need to steer in quad 4 to get there
so cos T = .417
T = 65.4 deg below x axis
that is south 25 deg east

go back and get
total ground speed North (y) = 550 sin T -7.81
total ground speed East (x) = 550 cos T - 44.3
now that you know T