The velocity of an object only really has meaning when it is
expressed relative to a set of axes or 'frame of reference'.

We usually consider the motion of
objects moving relative to the Earth, but since the Earth is orbiting the Sun and the Sun is
moving through the Galaxy this is a purely arbitrary concept. When dealing with more than one
body in motion the problem is complex - consider the difficulties in sending the space probe
Giotto to rendezvous with Halley's comet many millions of miles away!

We should always
specify the frame of reference relative to which an object is moving.

Think about the case where both bodies are
moving relative to a third. A simple example would be where the motions of both bodies are in
the same straight line - for instance, two cars travelling along a motorway. If both cars are
travelling in the same direction, one at 25 ms^{-1} and the other at 35 ms^{-1}
then their relative velocity is 10 ms^{-1} (by vector addition).

If they are moving
in opposite directions, however, the relative velocity of one car with respect to the other is
therefore 60ms-1 (See Figure 1).

What we are effectively doing is considering one car to be at rest and finding
the velocity of the other car in that frame of reference. To do this we must add the negative of
the velocity of one car to both cars' velocities. This effectively brings one to rest and we then
consider the velocity of the other car relative to it.

The situation is a little more complex
when the motion of the two objects is not in the same straight line. Consider the case shown in
Figure 2.

Here the two cars are still moving at the same speeds but this time at right
angles to each other as they approach a junction.

The relative velocity is found by using
a vector triangle.

If we add the negative of the velocity of the red car (-25) to both
the velocity of the red car and the green car we can imagine the red car to be at rest and then
find the velocity of the green car relative to it. (See Figure 3).

This is most easily done by drawing the vector diagram. The
resultant shown by the vector R is the velocity of the green car relative to the red car. In our
example the velocity of the green car relative to the red car is 43 ms^{-1} and in a
direction of 325^{ o}.

Think about the result and see if it matches your
common sense view of how you would see the green car moving if you were in the red car. It
would seem to be moving up the page and towards the left. It does actually fit the
result!

In fact we will not consider cars at all but the two aeroplanes shown in Figure 4 (a).

To find the velocity of plane A relative to plane B we must 'make plane B stationary'. We do this by adding the negative of plane B's velocity to both planes. If we were on plane B it would be like making that plane stop and we would then see plane A moving relative to us. This is done in Figure 4(b).