Relative Speed 

The diagrams below show two cars moving along the same straight road. 



The stated speeds are measured relative to point O which is a fixed
point on the ground. 



Where necessary, we will use the following notation: v_{A(O)} means
speed of body A relative to point O 



So, here we have v_{A(O)} = 40ms^{1} and v_{B(O)} = 30ms^{1}




Knowing these two figures, what is the speed of car A relative to car
B? 



In one second, A moves 40m along the road but B
only moves 30m. 

So, in one second, A moves further away from B in the positive sense. 

Obviously, if we imagine our self to be in car A, we will see car B moving
further away in the negative sense. 

In other words: 

v_{A(B)} = +10ms^{1} and v_{B(A)} = 10ms^{1}




This simple example illustrates that, in general, to find
the relative speed of two bodies, we subtract their speeds relative to a
third body. 

v_{B(A)} = v_{B(O)}  v_{A(O)} =
30  40 = 10ms^{1} 

and 

v_{A(B)} = v_{A(O)}  v_{B(O)} =
40  30 = 10ms^{1} 



Now
consider a slightly more dangerous situation: 



Again, the speeds stated are relative to O (in other words, measured in a frame
of reference fixed relative to the earth). 

Suppose we want to know the speed of car A relative to
car B. 

v_{A(B)} = v_{A(O)}  v_{B(O)} =
50  80 = 130ms^{1} 

which means, of course, that v_{B(A)} = +130ms^{1}




By the way, in case this all seems pretty obvious and intuitive,
be warned...
when Einstein looked at this more closely, he found it to need
significant modification when
the relative speeds involved are very great (see the
Relativity sections). 
