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)

*EQ8

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

*EQ9

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.

*EQ10

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.

Go back

This document was created on 23 August 1995
last modified on 23 August 1995
and is written by and copyright to btaylor@taylormade.com.au