Relative Velocities

Suppose there are two observers, one stationary, named Oo the other moving at v in the x direction relative to Oo, named Ov. By previous arrangement, when Oo and Ov are at the same position, a light pulse leaves a point Z. Oo and Ov each receive the pulse after times of t(0) and t(v) respectively, both at an approach speed of c in accordance to principle 1. Oo and Ov calculate the co-ordinates of Z relative to themselves at these times as (x(0),y(0)) and (x(v),y(v)) respectively.

Hence x(0) = ct(0) and x(v) = ct(v)


or t(0) = x(0)/c and t(v) = x(v)/c

In this time Oo sees Ov move a distance vt(0) and Ov is now x(v) units from Z which is seen by Oo as x(v).L(v) by *EQ6

x(0) = vt(0) + x(v)L(v) or


The observers now have formulae for calculating the position from Z from each others perspective.

Also since there is no motion in the y direction y(0) = y(v), substituting *EQ8 into *EQ9

which becomes

Next consider that the point Z is moving with velocity Ux(0) and Ux(v) in the x direction, and Uy(0) and Uy(v) in the y direction, relative to Oo and Ov respectively.

that is

How are the velocities of Z related? In Newtonian Physics

Ux(v) = Ux(0) v

but now things are different.


From *EQ9

the inverse of

NOTE: In the y direction Ux(0) = dx(0)/dt(0) = 0. Also, since v is constant, L(v) is a constant with respect to time.

Hence *EQ10 becomes

and Uy(v) = Uy(0).L(v) *EQ11

This result indicates that objects externally observed as approaching at a combined speed in excess of c will appear to each other to be approaching at less than c. In particular if the approach speed Ux(0) = -c, then Ux(v) also is c, i.e. principle 1.

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This document was created on 23 August 1995
last modified on 23 August 1995
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