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Suppose we toss a ball 2 meters up and it returns to the hand in .9 seconds, if we do the same in a moving train(constant speed), will it return at the same duration? .9 seconds? Is there an equation to prove this?

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    $\begingroup$ Basic premise is that all physics laws acts the same in all inertial reference frames, so ball time in air equation $$t={{\sqrt{u^2-2\,g\,h}+u}\over{g}} + {\sqrt{{2h}\over{g}}}$$ is the same for a ball tossed from the ground or from the inside of train or in any other inertial reference frame. $\endgroup$ Sep 2, 2022 at 9:01

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Under ideal (non-realistic) conditions, yes, the vertical motion is independent of any simultaneous horizontal motion. This is called the superposition principle and it applies to all geometric/Euclidian vectors, also i.e. forces, momenta, impulses and the like. It can be verified in experiments.

This principle lays for basis for the mathematical idea of splitting physical vector quantities into their components and then treating those components separately after which you can add up or collect the results into a resulting vector.

Realistically, air drag and other factors might cause the duration of your throw to not stay the same in both scenarios if it was real life. The superposition principle still holds true, but many factors that are not easy to control and prevent will influence the fall. Air drag varies with the speed (velocity magnitude) of the ball, and the speed in a simple vertical throw is smaller than in an angled throw, which would be the case with an initial sideways motion.

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When this type of question is posed, it is usually implied the train is enclosed, thus there is no external aerodynamic forces acting horizontally on the ball.

From a typical "high school physics class" perspective, the ball returns precisely to hand because that illustration is trying to prove a point about vectors and energy conservation... But excludes tiny factors that are beyond the scope of the illustration, which is only meant as a building block for later concepts. But at the engineering level, you have Coriolis and Eötvös effects that need to be considered. Coriolis says that the ball actually lands in your hand thousandths of an inch closer to the rear of the train than the front. And Eötvös says that the moving train ball will still land in your hand, but thousandths of a second earlier or later than the stationary throw.

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  • $\begingroup$ Neither Coriolis force, nor Eötvös effect are related to the train inertial reference frame, but to the rotating system instead (Earth), so technically these effects are irrelevant here in this question. $\endgroup$ Sep 2, 2022 at 7:23
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    $\begingroup$ @AgniusVasiliauskas This is absolutely incorrect. Neither the train, nor the people in the train are moving under inertial forces in an inertial frame. The train is rolling along the surface of a curved earth that itself is also rotating, and thus by definition the train and its occupants must have non-inertial forces acting on them. If the train is solely an inertial body, it would not be able to follow the curvature of the earth. This can be easily illustrated just in this thought experiment: $\endgroup$
    – Max R
    Sep 2, 2022 at 15:21
  • $\begingroup$ -cont: A train on a perfectly flat surface, non-rotating earth, is inertial. But that train enters a curve to the right. The train, and the people start moving to the right and untethered objects appear to move to the left. The passengers perceive they are in an inertial frame. So the ball’s movement to the left looks to them like a mysterious force has acted on it. No, they are the ones being acted on by the centripetal force of the rails, and the ball in flight is not, it is inertial. We are not the ones in the inertial frame. As humans on earth we are always in the rotating frame. $\endgroup$
    – Max R
    Sep 2, 2022 at 15:28
  • $\begingroup$ Yeah, then add Earth rotation around Sun, then yet add- Sun rotation around Milky Way galaxy core, and you may never know, maybe our galaxy/Local group is even rotating about something bigger...or universe is not flat and what not... What I wanted to say that science doesn't works that way,- messing everything into one pot and stirring up. It works another way,- abstracting many small details out and leaving just core principles. You can't describe reality in one go, it can be done in parts only. If Newton would be thinking as you - he could never derive his laws (and many scientists after). $\endgroup$ Sep 2, 2022 at 15:32
  • $\begingroup$ In middle school science we exclude even drag to illustrate kinematic concepts. In high school we begin to include things like drag but exclude Magnus. But in any college engineering class, you absolutely cannot simply refer to the train as the inertial frame. You would never be able to pass an intro level physics or engineering course if you referred to a train on the earth as an inertial frame. I understand it is your personal preference to visualize the problem only in the scope of the train as the inertial frame. Your preference has no bearing on the correctness of the answer. $\endgroup$
    – Max R
    Sep 2, 2022 at 16:09

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