R1rel

1. Are there nonlinear symmetries in Newton's equations?
2. Show that, in the case of a [math]\displaystyle{ k/|{\overrightarrow r}| }[/math] potential, there is an extra conserved quantity: the Laplace–Runge–Lenz vector [math]\displaystyle{ {\overrightarrow A}={\overrightarrow P}\times {\overrightarrow L}-mk{{\overrightarrow r}\over |\overrightarrow r|} }[/math].
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 [math]\displaystyle{ v/c=tanh(\chi) }[/math] and show that the [math]\displaystyle{ \chi }[/math]'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.