3 MOTION IN 3 DIMENSIONS 3.12 Relative velocity(^)
Figure 18: Relative velocity
to the ground. What is the vector velocity vg of the plane with respect to the
ground? In principle, the answer to this question is very simple:
vg = va + u. (3.52)
In other words, the velocity of the plane with respect to the ground is the vector
sum of the plane’s velocity relative to the air and the air’s velocity relative to the
ground. See Fig. 18. Note that, in general, vg is parallel to neither va nor u. Let
us now consider how we might implement Eq. (3.52) in practice.
As always, our first task is to set up a suitable Cartesian coordinate system. A
convenient system for dealing with 2-dimensional motion parallel to the Earth’s
surface is illustrated in Fig. 19. The x-axis points northward, whereas the y-axis
points eastward. In this coordinate system, it is conventional to specify a vector r
in term of its magnitude, r, and its compass bearing, φ. As illustrated in Fig. 20 , a
compass bearing is the angle subtended between the direction of a vector and the
direction to the North pole: i.e., the x-direction. By convention, compass bearings
run from 0 ◦ to 360 ◦. Furthermore, the compass bearings of North, East, South,
and West are 0 ◦, 90 ◦, 180 ◦, and 270 ◦, respectively.
According to Fig. 20 , the components of a general vector r, whose magnitude
is r and whose compass bearing is φ, are simply
r = (x, y) = (r cos φ, r sin φ). (3.53)
Note that we have suppressed the z-component of r (which is zero), for ease of
notation. Although, strictly speaking, Fig. 20 only justifies the above expression
for φ in the range 0 ◦ to 90 ◦, it turns out that this expression is generally valid:
i.e., it is valid for φ in the full range 0 ◦ to 360 ◦.
vg u^
va