In this vid why does the shadow trace a straight line instead of some curve ?
Intuitively it makes sense because at morning the shadow falls to west and at evening the shadow falls to east. He is basically connecting the shadow tips at these two times and claims that is the East-West line.
What I'm not able to convince myself is why the shadow tip traces a straight line at all times between morning and evening ? Is there any way to prove this using vectors or some other means ? Thanks!
(This is mostly just an amplification of @Henry's comment, which I didbn't read carefully until after I'd written it...sigh.)
The shadow tip does not trace a straight line. But at a local scale, it probably comes close enough to work for the purpose shown.
Here's the proof that it doesn't: move north of the arctic circle during midsummer, where the sun's above the horizon all day long, and appears, over the course of a day, to trace a circle around the whole horizon (rising and falling a bit as it does so). The arc traced out by the shadow-tip must then be a simple closed curve, hence not a line.
Now maybe you think "Well, that's a special case...if you're south of the arctic circle, you're OK. If you move a little bit south of the arctic circle, then things are fine." But the shape of the shadow-tip-trace must varies "continuously" as a function of latitude (**), so if you move just south of the arctic circle, you get an almost closed curve, hence it's once again not a straight line.
(**) I grant that this assertion needs proof, but not a lot: the displacement of the location of the shadow-tip from the stick-base, at a particular moment in the day, pretty clearly varies continuously as a function of the stick-base location (at least when there actually is a shadow at that hour); from that, and a bit of a compactness argument, you can make a pretty good case that the tip-trace can't jump from being a closed curve to being a straight line as you cross some particular latitude.