1) I wonder how many people on this list remember the
fully relativistic equation for angular aberration (as
in the aberration of starlight)?
I reckon some do, but many don't.
2) More importantly: How many people could rederive
it, quickly, with complete confidence in the answer?
3) Even more importantly, consider the general case,
where you can assume neither |v| << c nor |v| = c.
How are you going to handle that?
Hint: The answer to question (3) is exceedingly simple
and easy to remember, if you think about it in terms of
spacetime ... and it includes the answer to question (2)
as a special case.
To say the same thing the other way, Einstein (1905)
did tremendously more work to obtain a result that
is less general and harder to remember.
I don't remember the formula per se, because I can
rederive whenever I need it. I remember the idea for
rederiving it, because the same idea gets used over and
over again: Special relativity is just the geometry and
trigonometry of spacetime.
==============
Similarly:
There was a time, not so very long ago, when nobody had ever
seen any antiprotons, and certain folks were highly motivated
to build an accelerator to make some.
The question for today is, how much energy must such an accelerator
impart to the particles?
4) More specifically: Can you figure it out in your head,
quickly and easily?
I'll give you general idea: a beam of accelerated protons
smashes into a target containing a high density of stationary
protons. The reaction equation is:
p + p --> p + p + p^+ + p
I'll let you use a pointed stick, so you can draw diagrams
in the sand ... but I suggest that if you start writing down
numbers and/or equations you're doing it the hard way.
So, can you tell me, to the nearest round number, the design
energy for the accelerator? If you think about it the right
way, you can do it in your head, in less time than it takes
to talk about it.