# R1rel

From IFPA wiki
1. Are there nonlinear symmetries in Newton's equations?
2. Show that, in the case of a $k/|{\overrightarrow r}|$ potential, there is an extra conserved quantity: the Laplace–Runge–Lenz vector ${\overrightarrow A}={\overrightarrow P}\times {\overrightarrow L}-mk{{\overrightarrow r}\over |\overrightarrow r|}$.
3. A spaceship travels to a star 8 light-years away, in a time its crue considers to be 8 years. What speed does the ship travel at?
4. Frame S is at rest, frame S' has velocity v' w.r.t. S, and frame S" has velocity v" w.r.t. S'. What is the velocity of S" w.r.t. S'? Work this out using hyperbolic coordinates $v/c=tanh(\chi)$ and show that the $\chi$'s are additive.
5. In a distant galaxy, hydrogen emits microwaves of wavelength 21 cm. These are observed on Earth to have a frequency of 1.2 GHz. How fast is the galaxy receding from Earth?
6. A rod of length L moves with speed v along the x' axis. In its rest frame, the rod makes an angle θ' with respect to the x' axis.
1. Determine the length of the rod as measured by a stationary observer.
2. Determine the angle θ that the rod makes with the x axis, assumed parallel to the x' axis, in the stationary frame.

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