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Module 27: Lorentz Transformation: As an Exotic Rotation—Problem 1
In this problem, you will derive another simplifying feature of working with the rapidity, $\phi$ instead of the velocity $v$. It relies on simple hyperbolic trigonometric identities that mirror those of the ordinary trig functions.
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Question 1 of 4
1. Question
Consider the quantity $\sinh (\phi_1)\cosh (\phi_2) + \cosh (\phi_1)\sinh (\phi_2)$. By expressing each term as an exponential ($\sinh (\phi)=(e^{\phi} - e^{-\phi})/2$, $\cosh (\phi)=(e^{\phi} + e^{-\phi})/2$), simplify the quantity to one of the following four results.
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Question 2 of 4
2. Question
Consider the quantity $\cosh (\phi_1)\cosh (\phi_2) + \sinh (\phi_1)\sinh (\phi_2)$. By expressing each term as an exponential, simplify the quantity to one of the following four results.
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Question 3 of 4
3. Question
Consider $\tanh (\phi_1 + \phi_2)={\sinh (\phi_1 + \phi_2) \over \cosh (\phi_1 + \phi_2)}$. Using the results of the previous parts of this problem, express this in terms of hyperbolic trigonometric functions that depend only on $\phi_1$ or $\phi_2$ separately.
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Question 4 of 4
4. Question
Now for physics: If a rocket is moving northward with rapidity $\phi_v$ relative to you, and it fires a projectile in the northward direction with rapidity $\phi_w$ (relative to the rocket), what is the rapidity of the projectile relative to you?
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