If I stood on the equator and threw a stone in direction of rotation, how much farther would the stone go than if I threw it the other direction?

Assuming a spinning Earth of 6378.1 km (3963 mi) radius, how much farther would the additional momentum carry a stone if I threw it in the direction of the Earth's rotation (i.e. eastwards) when standing on the equator at sea level, than if I threw it with the same force the other direction? How much is the difference in percent or permille?

You, the stone, all locations on the equator, and the atmosphere a little above the equator are all moving with basically the same momentum initially. When you throw it, in the big picture, it gains total momentum (if throwing eastward) or loses total momentum (if throwing westward). But because its course and momentum do not shift it from being along the equator, its altered momentum will not have any "carryon" momentum difference horizontally that imparts apparent motion. In other words, for the east/west aspect DIRECTLY, the Earth's rotation will have no impact, it will continue doing what it was doing. You can consider its initial momentum from the Earth and its added momentum from the throw entirely separately... it will hit where you expect it to hit from this factor. The eastward or westward component have no bearing at all because it remains at the Equator.

It's like being on a moving train, and tossing a rock to your friend within the train car. Everything is moving the same rate, there is no difference.
So directly (from the eastward or westward direct of the throw), there is no difference at all along the equator.

However, there is still one difference in rotation rates for paths at the Equator. As the rock goes UP, it will have less velocity than required at that height (because the horizontal velocity needed to maintain the same rotation rate is more), and so it will lag behind. It will wind up west of where it should have... regardless of whether you threw it east or west. This difference is only a factor of the vertical component of the throw.

That said, if you're throwing it, it's going to be a pretty quick trajectory. Short distances and times have very little Coriolis deflection.

The amount of deflection becomes a nonlinear PDE (as early Coriolis alteration in the path changes both its subsequent landing time and its upcoming Coriolis deflection), so I don't believe you can derive a simple formula for the true percentage change.

But for a typical throw attempting an east or west motion, you won't have much vertical velocity, and gravity is very good at removing that component quickly. The deflection for a typical strong human's throw will be on the order of millimeters or less (this is true wherever you throw it on Earth).

The percentage of horizontal deflection would actually be infinite if thrown straight up (since a millimeter divided by 0 is still a "giant" change), and will quickly become a minuscule percentage if any horizontal component is given to the throw; a footballer's pass will not be noticeably altered by Coriolis anywhere on Earth. Air resistance/wind will be much much more significant factors. Even for missiles, which have much more velocity and so can fly for much longer times and thereby reach areas with significantly different rotational momentum, Coriolis deflection will only be on the order of a small percent. But a few percent can mean a few km/miles for intercontinental trajectories... kind of important for a targeted missile!

Only things that can stay "aloft" "permanently"... i.e. fluids (gases, and also liquids)... does Coriolis have a relatively giant impact (causing air that should otherwise flow directly from high to low pressure into air that rotates along isobars)

• Has the Coriolis deflection on Earth been purposefully demonstrated such as launching a missile from equator proximity (e.g. from Kourou) and launching the same missile at the same angle from north/south pole proximity (e.g. Esrange) to demonstrate they have differently shaped trajectories? Commented Sep 3, 2022 at 13:52
• I can't imagine such an event just to demonstrate Coriolis is worthwhile... every longrange missile launch must have Coriolis taken into the calculations, so every successful missile path would be evidence of such trajectories. And as noted, the difference in trajectories for paths with a short travel time (missiles included) would be very small (albeit very important for hitting a target). So no, I don't think any intentional demonstration like that has been done, and not sure it would be of any real use. Commented Sep 3, 2022 at 14:11
• Its use would be proof of either the rotation of the Earth around its axis, or of the centripetal force of the universe onto Earth (as both would have the same effect). Commented Sep 3, 2022 at 15:44
• Proof to who? People who use the formulas on a daily basis, mathematicians, etc don't need any further proof. And those who you're suggesting still need proof of the rotation of the Earth... must already not take anything existing as evidence, concocting some conspiracy explanation... so they surely wouldn't accept any such missile experiment either? And not sure what you're even suggesting by "centripetal force of the universe onto Earth"??? Commented Sep 3, 2022 at 19:16
• I meant that if the Earth stood still and the universe revolved around it, this would have the same effects as if the Earth rotated around its axis, e.g. that geostationary satellites don't fall down. It's a matter of your reference point of view. You're correct that such missile experiment wouldn't be accepted by say Flat Earthers because they otoh would claim this would be due to a dome rotating above the Earth. Still, it would be a nice and interesting experiment. Time dilation didn't need proof either but was demonstrated by taking a clock on a plane that circumnavigated the Earth. Commented Sep 4, 2022 at 5:53