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Gravity Probe B

Testing Einstein's Universe

Special & General Relativity Questions and Answers

If two objects approach at 75 percent the speed of light, why are they not traveling 1.5 times the speed of light?

The special relativistic formula for adding velocities to determine a relative velocity between approaching or receding bodies is:

            v1  +   v2
V  =  -----------------------
                 v1 x v2
        1   +    --------

so that if v1 = 0.75c and v2 = 0.75c then
            0.75  + 0.75
V  =   ------------------------   c
                0.75 x 0.75
         1    + -------------
or V = 0.96c. This means that if you were sitting on either body 1 or body 2 using a clock and a meter stick at rest with respect to you, ( called the 'proper measuring system'), you would note that the relative speeds were 0.96 times the speed of light. If you were sitting on a third body watching these two approach each other using a third reference system, if you did not properly allow for the difference in the time dilation factor, you might be tempted to, say, divide the distance traveled by these two observers on bodies 1 and 2, by the wrong time reference, and end up with a velocity greater than light. The problem is that, because of the finite time delay introduced by the propagation of light, you will get confused in how to measure, simultaneously, when the two bodies arrive at a certain separation that you use to measure their velocity. This is all automatically included in special relativity which gives the correct predictions for all motions near the speed of light.

I am sorry if this explanation is a bit unsatesfying. Some problems that we all have in understanding how nature works, have more to do with our conviction that our intuitions are correct, despite the fact that our intuitions are often flawed in areas of experience outside of our normal environment.

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All answers are provided by Dr. Sten Odenwald (Raytheon STX) for the NASA Astronomy Cafe, part of the NASA Education and Public Outreach program.