Relative Motion

at Very High Speeds

As a young child, I was fascinated by the following question that my father posed to me: If a machine gun, which fires bullets at 100 miles per hour, is mounted on an airplane that flies 200 miles per hour, will the airplane run into its own bullets?

Now equipped with our ideas and equations of relative motion, we know that if the bullets have a speed of 100 mi/h relative to the airplane, they will have a speed of 300 mi/h relative to Earth and, indeed, the airplane will not run into its own bullets.

But how do we answer the following question which seems quite similar?

If an alien spacecraft fires photon torpedoes forward with a speed of 0.75 c (with respect to the spacecraft, of course), while the spacecraft itself has a speed of 0.3 c with respect to Earth, how fast are the torpedoes observed to travel with respect to Earth?

The obvious, common-sense, intuitive answer is 1.05c; this is what the Galilean Transformations would give. But velocities greater than c do all sorts of terrible things to the Lorentz Transformations from Special Relativity (like asking for square roots of negaive numbers!).

The diagram below shows frames A and B with relative velocity v. In frame B, object C is observed to have a speed of vCB in the x-direction. What is the speed of C with respect to A, vCA?

The answer to this question will have to wait until we study Special Relativity. For very high speeds -- that is, for speeds near the speed of light -- many of our intutive expectations must be modified. This is part of the excitement of Modern Physics.

Relative Motion

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(c) 2005, Doug Davis; all rights reserved