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Question
I am interested in Special relativity and have noticed something i do not understand to do with velocity addition.If a spacecraft takes of from earth and is travelling at 3/4 the speed of light and launches a missile at a speed of 3/4 the speed of light in the same direction as the spacecraft. I used the relativity velocity addition formula W = (u + v)/(1 + uv/c^2) and this gives the speed of the missile as 24/25c (or 0.96c) if measured by an observer on earth. Now if the missile has a clock the special relativity time dilation equation say the the clock on the missile will be running at 16/25 slower. the clock on the spacecraft is running 4/5 slower to the observer on earth. An astronaut on the spacecraft will see the clock on the missile running 4/5 slower than the clock on the spacecraft. If the astronaut set their clock to run at the same speed as the clock on the missile and the observer on earth would then see the clock on the spacecraft as running (4/5)*(4/5) = 16/25 slower. I do not understand why this is not the same as observing the clock on the missile directly?Similarly the astronaut and the observer on earth would view the mass of the missile is different.

In such cases, when you observe a discrepancy, always remember that the fundamental tenet of relativity is that the laws of physics are the same for all observers.  That's why you start to suspect either your math or your interpretation in such cases.

In this particular case, you actually have to use the velocity addition formula to calculate the time dilation on the missile's clock, you can't just multiply two factors of "slower" together.  The slowdown factor for the missile relative to the spacecraft doesn't account properly for the observer on Earth because the relative velocity is no longer correct.  The only correct way to do it is to first compute the velocity relative to the observer on Earth and then to calculate the time dilation factor.

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#### Dr. Stephen O. Nelson

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I can answer most basic physics questions, physics questions about science fiction and everyday observations of physics, etc. I'm also usually good for science fair advice (I'm the regional science fair director). I do not answer homework problems. I will occasionally point out where a homework solution went wrong, though. I'm usually good at explaining odd observations that seem counterintuitive, energy science, nuclear physics, nuclear astrophysics, and alternative theories of physics are my specialties.

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Ph. D. from Duke University in physics, research in nuclear astrophysics reactions, gamma-ray astronomy technology, and advanced nuclear reactors.