### Question

How fast a person must be going in a car to make a red light appear green?

Asked by: Katherine

### Answer

Well, I would simply say 'relativistic speeds which
a car cannot achieve with the current technology',
and leave it at that. But, for the curious PhysLink
reader, let me actually find out _how_ fast:

This is simply the relativistic Doppler shift for light.
The following formula can be found in any textbook:

f' = f sqrt(1 + beta)/sqrt(1-beta)

where f is the frequency of the light for an observer
at rest with respect to the light source, f' is the the
frequency of the light for an observer approaching the
source with velocity v, and beta is just v/c, where c
is the speed of light.

Note that this equation does not come out of the blue. It
can be derived from the Lorentz transformation for the
four-momentum of a photon, (as well as otherwise) but
I will not go into that here.

First, we need some data, or 'trivia'. The frequency of
red light is about 4.5x10^{14} Hz, and the frequency
of green light is about 5.5x10^{14} Hz.

Second, we need to express beta in the above equation
in terms of f and f'. This takes a few steps of algebra,
but finally we get the following equation:

beta = {1 - (f/f')^{2}} / {1 + (f/f')^{2}}

Plugging in 4.5x10^{14} for f, and 5.5x10^{14} for f',
we arrive at the desired answer:

beta = 0.198

which means

v = 0.198 c

... or about one fifth of the speed of light.
Considering the speed of light is about 300,000
kilometers per _second_ or 186,000 miles per _second_,
that's pretty darn fast. Calculating the exact figures
is left as an exercise for the PhysLink reader.

Answered by: Yasar Safkan, Ph.D. M.I.T., Software Engineer, Istanbul, Turkey
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