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Relativity
Measuring
Relative Velocity
How can we measure the relative
velocity of two bodies when the distance between them is very
large? Whatever method we choose, it will involve the use of
light or some other electro-magnetic radiation. Consider two
observers, A and B, moving with relative
velocity v as shown below. Notice that the diagram is drawn
from A’s point of view.
A sends pulses of light
to B at intervals of T seconds (as measured on A's
clock). If A and B are moving away from
each other, as shown in the diagram, B receives the
pulses at longer intervals because each pulse has
further to go than the preceding pulse. To help in
measuring the relative velocity, we will define the constant,
k as follows
B has a mirror which
reflects the pulses back to A. We will consider the
following sequence of events
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1. |
At
the instant when A and B are together,
they set their clocks to zero and the first pulse of
light is sent. In other words, the first pulse is sent
and received at t = 0. |
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2. |
The
second pulse, sent by A at time T is received
by B at time kT. |
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3. |
The
reflected pulse is therefore received by A at
time k2T.
Now, according to A,
the time at which the reflection of the second pulse
occurred must be half way between the time of
sending the pulse and the time of receiving its
reflection, that is |
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(T + k2T)/2 |
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4. |
This
means that the light went from A to B
and back in a time given by |
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k2T
- T |
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Therefore
the reflection occurred at the instant when the
distance between A and B was |
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(k2T
- T)c/2 |
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So,
if the velocity of B relative to A is v,
we have |
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 |
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and
this gives us a method of measuring relative
velocities. |
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