Years ago, the textbook I used for Intermediate Algebra mentioned relativity in the rational functions chapter. They gave an expression for observed speed (of galaxies separating). I wanted to help my students practice the algebraic steps involved in solving an equation (with lots of variables) for one variable, and made up a silly story about a galacto-cop chasing a possibly speeding pirate. I have a few questions.
The equation I used was equivalent to s = (u+v) / (1 + uv/c2), where u and v are the observed velocities of two objects moving in opposite directions. s is the speed at which they're separating. At earthly speeds, s would equal u+v. Simple. But near the speed of light things get complicated. If u and v are each over half the speed of light, u+v would give us an s value over the speed of light. That's apparently not possible. Einstein (and others?) came up with the equation above, which describes relativistic effects on velocity. I think. (Please correct my statements here if they're inaccurate, misleading, or confusing.)
What I wanted to do with my students was to solve for u. I got u = (s-v) * c^2 / (c^2 - sv). First question, is this legitimate?
My story was that the galacto-cop's 'radar' (what else should I call it?) gave her the pirate's speed of separation from her ship, but she wants to know the pirate's 'true' speed, and has to figure this version of the formula out. I claimed the galactic 'speed limit' was 1/4*c.
My understanding of relativity is weak, and I'd like to get this story down a little better before I tell it in class this semester. Help?