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- Are there nonlinear symmetries in Newton's equations?
- Show that, in the case of a [math]k/|{\overrightarrow r}|[/math] potential, there is an extra conserved quantity: the Laplace–Runge–Lenz vector [math]{\overrightarrow A}={\overrightarrow P}\times {\overrightarrow L}-mk{{\overrightarrow r}\over |\overrightarrow r|}[/math].
- 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?
- 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]v/c=tanh(\chi)[/math] and show that the [math]\chi[/math]'s are additive.
- 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?
- 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.
- Determine the length of the rod as measured by a stationary observer.
- Determine the angle θ that the rod makes with the x axis, assumed parallel to the x' axis, in the stationary frame.