A man walking at a rate of 4 miles per hour toward the forward car of a train when the latter is traveling at a speed of 50 miles per hour is actually traveling at the speed of 54 miles per hour over the round. His relative speed with respect to a fixed point on the moving train is 4 miles per hour. This simple concept of superposition of speeds can be extended readily to the case of differing directions making use of the reference system velocity and the relative velocity by use of vector addition.
Consider the two-dimensional displacements pictured in Fig. 2.10-1. The
axes indicated by the full lines represents a fixed
reference system in absolute space. The set of axes shown
by dashed lines represents a reference system which is moving
without rotation. Two positions of the origin O' of the
moving system corresponding to times 
and 
are shown in
Fig. 2.10-1(a). Points 
and 
correspond to the
positions occupied by a moving object at times
and , respectively. These positions can be represented
by the radius vectors 
and 
in absolute space, or by the radius vectors
and 
relative to the translating
coordinate system. The actual displacement of the object in
absolute space is 
. This displacement which would
be observed if one traveled with the moving coordinate system
is shown in Fig. 2.10-1(b) as 
which
represents 
. The latter
figure is obtained by merely transferring the relative
vectors and to the common
relative reference point O'.
If 
and 
are the radius
vectors of the origin of the relative reference system at
times and , respectively, then it can be seen
from Fig. 2.10-1a that
Consequently,
or
The vector addition is shown in Fig. 2.10-1b.
If we divide Eq. (95) by 
and take the
limit as approaches zero we obtain
or
where
is the absolute velocity of the object,
is the translatory velocity of the moving coordinate system, and
is the relative velocity of the object referred to the translating coordinate system. Eq. (97) applies equally well for three dimensional motion.
If the relative coordinate system undergoes rotation as well as translation then another term must be added to Eq. (97). The effect of rotation of the relative coordinate system can be analyzed separately from the effect of rotation. Consider the two-dimensional picture shown in Fig. 2.-10.2. The origin of the relative system is taken at the same point as the absolute system
(full line coordinates). However, the relative system has rotated
through an angle 
about the z-axis at time
and 
at time . The lengths of the vectors
and are invariant between the two
systems but the polar angles 
differ. The angles
and 
represent the angles which the
radius vectors and , respectively,
form relative to the x axis, while 
and 
refer to the angles of these radius vectors with respect to the
relative coordinate axis x'. Thus
Let 
represent the angular velocity of the relative
coordinate system. This represents a vector along the positive
z axis whose magnitude 
is the counterclockwise
rate of turning, i.e.
and the direction of follows the usual right handed
screw convention. If we regard the angular displacement
as small (corresponding to a
small time interval then a fixed
point P' in the relative coordinate sytem will actually
undergo an absolute displacement
where 
is the absolute radius vector of the point. This is shown
graphically in Fig. 2.10-3. If we regard 
as the mean radius
vector for the period to
then we can write approximately
This vector addition is illustrated in Fig. 2.10-2b. Dividing
by and taking the limit gives the exact relation
or
for pure translation and rotation of the relative coordinate system.
If the relative coordinate system undergoes both translation and
rotation and 
is interpreted as the relative velocity
referred to this system then the absolute velocity is
where 
is the velocity of translation and
the velocity of rotation of the relative coordinate system as
previously introduced. The combined situation of translation
and rotation of a two-dimensional coordinate system is illustrated
in Fig. 2.10-4.
In the case of three dimensions, Eq. (104) is still applicable.
However, the angular velocity can now take on any
orientation in a direction relative to the coordinates,
just as , 
and .