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)?
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
To answer these questions, we need to calculate the heading and ground speed of the plane. Let's go step by step:
Step 1: Calculate the component of the wind along the heading of the plane.
The wind is blowing from N80oE, which is 80 degrees to the east from the north. We want to calculate the component of the wind along the heading of the plane to determine its effect on the plane's motion.
To do this, we need to decompose the wind vector into its north (N) and east (E) components.
North Component (N):
N = wind speed * cos(angle between wind direction and north)
N = 45 km/h * cos(80o)
N ≈ -10.3 km/h (negative because it is acting southward)
East Component (E):
E = wind speed * sin(angle between wind direction and north)
E = 45 km/h * sin(80o)
E ≈ 43.8 km/h (positive because it is acting eastward)
So, the component of the wind along the heading is approximately 43.8 km/h in the east direction.
Step 2: Calculate the heading of the plane.
The heading of the plane is the direction the pilot should steer the plane in order to cancel out the effect of the wind and reach the desired S20oE direction.
To calculate the heading, we need to find the angle between the desired direction and the east direction. The desired direction is S20oE.
East of South angle (ES):
ES = 90o - 20o
ES = 70o
To determine the heading, we need to add the angle between the east direction and the component of wind along the heading (ES) to the east direction.
Heading = N80oE + ES
Heading = 80o + 70o
Heading ≈ 150o
So, the pilot should steer the plane to a heading of approximately 150 degrees.
Step 3: Calculate the ground speed of the plane.
The ground speed of the plane is the combination of the aircraft's airspeed and the wind's effect on its motion. We can use the Pythagorean theorem to calculate the ground speed.
Ground speed = √(airspeed^2 + (wind component along the heading)^2)
Ground speed = √(550 km/h)^2 + (43.8 km/h)^2)
Ground speed ≈ √302500 km^2/h^2 + 1916.44 km^2/h^2
Ground speed ≈ √304416.44 km^2/h^2
Ground speed ≈ 551.8 km/h (rounded to one decimal place)
So, the actual ground speed of the plane will be approximately 551.8 km/h.