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How can a person trapped in a box figure out, without looking outside, whether the box is stationary or moving with a constant velocity?

Is it even possible, considering the fact that the box is an inertial frame of reference? (since acceleration, and thus net force, is zero)

Additional question: Assume that the box is moving with a constant velocity. When a ball is thrown at a wall, will the time taken by the ball to reach the wall be any different from the time taken when the box is stationary? (If I picture myself in a closed room, I don't see why this should be possible - yet I can't convince myself that this is the case. If the wall is moving away from the ball, it seems reasonable that the time taken should be more. Likewise, it feels reasonable to think that the ball should take less time to reach the wall that is moving towards it.)

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The last two sentences in your question imply that the box is moving in a horizontal direction.

In the cases of the person either facing or with his back to the leading wall, throwing the tennis wall at either wall will take exactly the same time as if the box was stationary.

The velocity of the throwing hand with reference to the outside world will exactly mirror the velocity of the wall it is thrown at. The result of that is, though the hand, for one example, may impart extra velocity to the ball, with reference to the outside world, therefore the ball may be seen to be going faster by someone outside the box who has x-ray vision, the wall it is being thrown at is moving away at the exact same rate as the "extra" velocity of the ball. The ball will therefore hit the wall at exactly the same time as if the box was not moving.

The result is the same for the opposite direction, when to an outside observer the ball may be seen to be moving "slower", but then the wall it is beiong thrown at is moving towards the ball at the exact rate that the ball has been "slowed". Therefore, the time remains the same as if the box was not moving.

If the ball is thrown at either "sidewall" in this horizontally moving box, the result is still the same as if the box was not moving. I refer you to the laws of inertia, or Sir Isaac Newton if you prefer.

Intriguing question. I had to work my way through possible "traps" before I answered it.

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  • $\begingroup$ Conclusion: the person inside the box can never know if the box is stationary or moving at a constant velocity. $\endgroup$
    – user171258
    Oct 6, 2017 at 14:43
  • $\begingroup$ Thank you! So is there any way a person in the box can figure out if the box is stationary or moving with a constant velocity? $\endgroup$
    – Art
    Oct 6, 2017 at 14:47
  • $\begingroup$ See below answer if you wish to account for special relativity. You can use the principles stated in this answer for general purposes, but remember that it becomes less and less accurate as velocity increases. $\endgroup$ Oct 6, 2017 at 14:50
  • $\begingroup$ Well, I admit it was the 1960s when I led my high school four years running in physics, general science and mathematics. However, from the point of view of the observer, I mean the observer in the box keeping time, throwing the ball and looking at the clock travelling at the same constant velocity with him, that time was perceived at the same rate as when he was still. Surely the outside observer, provided he or she was stationary compared to the moving box, would be the only one noticing the relativistic effects? And the question was asked from the perspective of the person in the box. $\endgroup$
    – user171258
    Oct 6, 2017 at 15:06
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    $\begingroup$ Yes, I think I didn't word my relativistic comment very diplomatically. Although Art did carry on to write in his additional question you quoted, that the observer could be himself in the room and that clarified to me anyway that his question was clearly to do with an "in-room" observer, I believe Kieran that you did Art a great service in pointing out that there was a broader perspective to his question. That he very much appreciated your answer is indicated in his heartfelt "Thank you!" to you. You saw the broader "question behind the question", which I did not. Thank you from me. $\endgroup$
    – user171258
    Oct 6, 2017 at 17:30
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The phenomenon of not being able to distinguish between inertial (non-accelerating) frames of reference is called the Equivalence Principle. You would not be able to tell that you were moving.

Throwing the Ball

Your reference frame

From your perspective inside the box, the ball would travel at the same speed towards all sides (excluding up and down if counting gravity) of the box.

External reference frame

If you throw the ball along your velocity vector, the velocity that an external viewer would see the ball moving at would can be calculated as:

$$u’ = \frac{v + u}{1 + \frac{vu}{c^2}}$$

Where

  • $v$ is your velocity
  • $u$ is the velocity of the ball measured by you
  • $u’$ is the velocity of the ball measured by the observer

You can see from that formula that the resulting velocities would not cause the ball to reach the front and back of the box at the same time if you had a non-zero velocity. This phenomenon is known as Relativity of Simultaneity: https://en.m.wikipedia.org/wiki/Relativity_of_simultaneity

In physics, the relativity of simultaneity is the concept that distant simultaneity – whether two spatially separated events occur at the same time – is not absolute, but depends on the observer's reference frame.

If you throw the ball perpendicular to your velocity vector, we can discount the relativistic velocity addition because you are unaccelerated in that direction.

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  • $\begingroup$ Thank you! Your answer was helpful. I'll look up relativity of simulataneity. $\endgroup$
    – Art
    Oct 6, 2017 at 14:51

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