Relative velocity( Classical and Relativistic)



Galilean or Classical relativity equation

vAE => Velocity of A with respect to Fixed Frame
vBE => Velocity of B with respect to Fixed Frame
vAB => velocity of A with respect to B

vAB=vAE – vBE
or
vAB=vAE + vEB

Let
vAE =10 m/s( towards positive x-axis)
vBE =6 m/s ( towards positive x-axis)

vAB=10-6 =4 m/s
This equation works well for normal velocities but it does not hold good for speed reaching speed of light

Relativistic Velocities
Suppose
vAE= 0.6c
vEB= 0.8 c
Using the classical relativity equation above, you get

vAB=vAE + vEB = 0.8c + 0.8c = 1.6c

In other words, this predicts that A will measure B’s velocity as 1.6 times the speed of light, which is forbidden by Einstein’s Special Theory of Relativity, which says, among other things, that the speed of light is the ultimate speed limit in the Universe.

In Special Relativity the classical equation is modified to be:

v_{AB}=\frac{v_{AE}+v_{EB}}{1+\frac{v_{AE}v_{EB}}{c^{2}}}

So Using Special Relativity, we get:

vAB=.98 c

In other words, A would measure B’s velocity as 98% of the speed of light.





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