I apologise for the volume. I got sidetracked by life this past week and have replied to all the pending posts in one horribly wrong epistle.
PJ
PJ> Consider the time and velocity of any object completing the circumnavigation of a point (e.g. center of The Earth). At any point in time that object has an X and a Y component of it's velocity.
I think there is a weakness here. I don’t know whether it invalidates the argument.
There is indeed an “X and Y component” of its velocity, but in what co-ordinate system?
For a low-velocity, short-duration flight (tossing a football in the air from my old hands) we can treat The Earth as a stable plane. But you will go on to treat The Earth as a rotating sphere, with the projectile’s initial reference at the time it leaves the barrel rapidly disappearing over the horizon.
I don’t want to get so far into maths that we use The Moon as an absolute point and calculate everything with “X and Y components” relative to the centre of The Moon.
PJ> ... the "X" portion (with respect to the center point) ...
Here too, if you are going to suggest cartesian vectors then I think you can’t have a “centre point”. You have to use polar coordinates (out of my depth soon) and radian measurements (now out of my depth)
Leif
Leif> ...it'll have to be done at either the North or South pole ...
Which I interpret to mean “we can safely ignore The Earth’s rotation”. A coward’s way out (grin).
Any solution should be accurate at the equator and at the pole and at all points in between, the poles being some sort of boundary condition.
AlanMiller
AlanMiller>... the ground rotates beneath the bullet ...
I’m getting out of my depth again. It is a fine point to argue that the ground rotates away from the bullet, rather than beneath it. “Away From” suggests a low-velocity low-altitude Flatlander scenario, whereas we are looking at what is probably a transition area between tossing a football and firing an ICBM.
PJ
... (their linear "X" velocities are the same) ...
My point above on Post 213751. I think that you are using two different frames of reference again, one cartesian and one polar.
It’s late at night, though, and my brain hungers for Bill Bryson and Garrison Keillor ...
I do trust all your calculations, if only because I didn’t specify any initial quantities.
I am having trouble visualizing The Earth rotating and the bullet rising, and their positions relative to each other.
I note that NASA doesn’t launch a rocket straight up away from The Earth. Well, only for the first fifteen seconds or so, then they roll it over onto its back and send it on a trans-atlantic tour. That is, rockets spiral up unto orbit.
Maybe we DO have to be Rocket Scientists!
PJ> Were I Issac Asimov writing one of his wonderful science essays that I ate up as a young lad, I might consider the effort. But for Chris, well ...
You CAN compare me to Asimov. He is like any one of his books. I am the mosquito squashed between page 71 and 72! (which is, of course, impossible ...)
AlanMiller
Alan> That sounds like a Coriolis-type correction to me.
I know that winds are above the plane of The Earth, but if I think of them slipping across the surface of The Earth, then from what I remember of Mr deKurloi’s geography lessons, the Coriolis effect on winds occurs when a wind moves with a component of its velocity towards or away from the poles.
That is, a body of air flowing from (roughly) new York to Florida will appear to veer westwards(?) relative to a straight line drawn from New York to Florida.
And you all know what I mean by a “straight line”, I mean a piece of string stretched taut between the two cities, curved to fit the shape of The Earth.
PJ
I wish your image didn’t look like winds creeping across the surface of The Earth instead of a bullet rising above The Earth. I’m trying to keep an open mind here.
PJ> ... is being applied in different reference frames so the end point location of the gun and the bullet will be different after a period of time.
This is so seductive. “different reference frames” appeals to me, but I still have a nagging feeling that the bullet is held by gravity and so is in an orbit-like trajectory. I know it is a parabola relative to The Earth’s surface, but it seems orbital with respect to a fixed point on The Earth’s surface.
(I must remember to sign this post “Confused” of Toronto)
PJ> The ball bearing will land eastward of the center of the base
I think I agree with this. You were saying in Post=213806 that the rifle, travelling upwards, falls behind the barrel, but the ball bearing, traveling downwards, falls ahead of the tower. That makes relative sense.
AlanMiller
Alan> ... so small that we can consider its surface to be a non-rotating, moving plane ...
Which is the case when I toss a football into the air. We can consider The Earth as an immobile plane. However this argument can not, surely, apply to an ICBM fired straight upwards (except for angling a bit for atmospheric variables). Admittedly you’d have to be an idiot to fire an ICBM straight up and hope that it would come back and visit your cafeteria, still ...
I can imagine (no atmosphere) firing a projectile upwards, at right angles to the sphere so that the sphere had time to rotate through ninety degrees before the projectile returned to the surface. At that scale my mind shrugs and supposes it can accept lateral differences.
Then I think about the LLM rising up to rejoin Apollo. Did that just shoot vertically off the surface of The Moon and trust/hope/calculate that the orbiting Apollo would appear over the horizon at just the right time and distance to effect a meeting? If so, shouldn’t we draft PJ to argue the travel in the opposite direction? I mean, the LLM is like the bullet going “straight up”.
PJ
PJ> So with that mind problem set up,...
I rather like this experiment. We ignore The Moon’s path around The Earth, and just consider The Moon as a rotating sphere.
This almost makes sense to me. I still have trouble visualising what the projectiles so-called horizontal component of velocity is wrt The Moon.
AlanMiller
Alan> ... it would mean a refresher course of my Lagrangian mechanics ...
And there, Gentlemen, I think I will leave you!
Thanks to all who contributed, especially PJ and Alan. I learned more than I had anticipated.
I’m also glad that we focussed on this Radial/Cartestian/Coriolis effect and not any other factor that I had failed to think of.
(signed) “Confused” of Toronto