Anonymous

A small airplane flies at 60 m/s relative to the air. The wind is blowing at 20 m/s to the south. The pilot heads this plane east.

(a) What is the speed, v, that the plane will travel relative to the ground?

v = m/s

(b) Relative to the ground the plane moves at an angle measured south of east. What is that angle (in degrees)?

angle = ° *

18.4 OK

(c) What direction must the pilot head his plane in order to travel due east relative to the ground? Give the angle in degrees measured north of east.

angle = °

(d) What will be the speed of the plane,v, relative to the ground with this new heading?

v = m/s

Relevance

a)

The x-component of the airplane velocity is 60m/s.

There is no y-component for the velocity of the plane or it is 0

The x-component of the wind = 0

The y-component of the wind is 20m/s

The resultant vector for the heading of the airplane is sqrt (60^2+20^2) = sqrt (3600 + 400) =sqrt (4000) = 63.24

b)

Inv tan theta=20/60 =18.4 degrees

The plane travels with a speed of 63.24 m/s heading 18.4 south of east.

c)

If you don't wish to give the required data to find your request, then you go for the solution as follows:

20 sin beta = X*sin theta

X = 20*sin beta/sin theta, where x is the velocity of the airplane north of east, and theta is the angle at which the airplane travels, and beta is the angle of the wind south of the east.

If you choose beta 30 degrees, and theta 40 then the velociy of the airplane = 15.62 40 degrees of the east.

However, in order for the airplane to travel due east, then the y component of the airplane heading and the y-component of the wind must be equal but in opposite direction. If the airplane y-component is equal to 60 sin Theta, the y-component of the wind must = -20sin beta

60 sin theta = 20 sin beta

Sin theta = x-component of plane velocity 60 m/s= x1/60

Sin beta of the wind must = -x2/20.

60*X1/60 =20*x2/20

60sin theta = 20 sin beta, sin beta = 3(sin theta)

Beta is the angle at which the wind blows south of East

Theta is the angle of the airplane heading north of east

What ever the angle of theta at which the plane is heading, beta must equal 3 times theta provided that the magnitude of the velocity of airplane = 60 m/s, and the magnitude of the wind velocity = 20 m/s

If these conditions are met, then the heading of the airplane will be due east.