Two observers are moving toward each other with some relative speed between them. One then throws a ball to the other when they are at some distance from each other. These relative speeds are not great, so we probably don't need to bring relativity into this. Now, from the point of view of the person throwing the ball, considering that one stationary, then, the ball has travelled the original distance the two were separated by when the ball was thrown minus the distance the other has moved closer since then until the ball is caught, right? But according to the one catching the ball, the original distance the ball was thrown from is the distance the ball has travelled, regardless of how close the thrower has moved since then, isn't it? Which one is right? What is the actual distance the ball has travelled?

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Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

They both get the same. They are both right.

Unless you have some weird definition of distance, ball, or right.

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0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0....

Only if the catcher stopped moving when the ball was thrown. If (s)he kept moving as you seemed to be saying (s)he would, then no, the distance the ball has to travel is less than the distance between pitcher and catcher was at the time the ball was thrown; the distance the ball travels is that amount minus the amount that the catcher moved after it was thrown.

grav, I don't know if is just me, but first you say both are moving toward each other and then you say...[Now, from the point of view of the person throwing the ball, considering that one stationary,]

Makes me wonder why you didn't just state it that way to begin with???

ETA: Ahhhhh, I see now grav, forget this post

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RussT
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Everything is, as it should be, otherwise, it wouldn't be!

Playing with ASCII "art"...

T1. X and Y are moving towards each other, each at speed 2. X has a ball.
T2. X throws the ball at speed 4.
T3. X and Y continue to move ("-") and the ball is now moving too (".").
T4. Y catches the ball.

Wouldn't both observe that the ball has travelled 8?

That is, the distance between X and Y at time T2, minus the distance travelled since then by Y?

(added later...)

If X considers themselves stationary, then they'd have to see the ball as having travelled at speed 2 [not 4] (and thus a distance of 4 [not 8]).

Is your point that their whole idea of time and distance would (on the stationary assumption) be different to Y's?

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Measure once. Cut twice. Power tools are fun.

Yes. I'm getting a different distance for the path of the ball, depending upon which observer we consider stationary. It seems too simple a scenario for this to be the case, but that's what I get so far.

__________________
Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

If you are getting different times of flight, it probably is because you did not keep the ball at the same velocity relative to the pitcher. BTW you did not explicitly specify the ball's velocity in the OP.

I am assuming you mean the same action, with the only difference being the choice of one inertial frame of reference or the other. Therefore I am assuming that the ball's velocity relative to the pitcher is fixed.

When analyzing as if the pitcher were stationary, it would be a slow pitch with the catcher charging. When analyzing as if the catcher were stationary, it would be a fastball with the pitcher chasing it. Either way we would get the same time of flight and the same final separation of the two players.

If a walking man were to throw a ball and, a second person were to catch it. The path of that balls flight does not change from any view point. Your perception of that flight will depend on your point of view. The actual path the ball has traveled is the same. Its like watching basketball.

Okay, let's say two observers are moving toward each other with a relative speed of 30 m/sec. One of them throws a ball toward the other with a relative speed of 10 m/sec to his own frame when the observers are at a distance of 20 meters from each other. That means that the catcher sees the ball travelling toward her at 40 m/sec.

Now, to the catcher, the ball was thrown when it was 20 meters away and travels that entire distance to her. She will see the thrower lag behind the ball, but it does not matter how the thrower travels after the ball is thrown. So to the catcher, the ball has travelled 20 meters at 40 m/sec, so takes 1/2 second to reach her.

From the perspective of the thrower, however, the ball was only thrown at 10 m/sec relative to him, but the catcher is also moving toward him at 30 m/sec. So during the time the ball is in flight, the ball travels 1/4 of the distance toward the catcher while the catcher travels the other 3/4 of the distance toward the ball. To him, then, the catcher has travelled 15 meters of the original 20 meters while the ball has only travelled 5 meters. Since he sees the ball travelling 5 meters at 10 m/sec, the time of travel is 1/2 second, the same as that for the catcher, but the distance travelled is quite different.

The question is, if all of these speeds are relative between observer and each observer and the ball, so that either can consider themself stationary and measure the same duration of flight of the ball, but over different distances travelled, then which one is right? What is the true distance the ball has travelled, 5 meters, 20 meters, or something else?

__________________
Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

And if you run alongside the ball, it will travel no distance at all in that half second, relative to you, because you move your own coordinate frame along with you as you run.
All the distances are correct, each in its own reference frame.

Grant Hutchison

That's the only way I see that it can be also. Measured distances must be purely relative. I knew that to be the case when considering a stationary system with an arbitrary point of origin, but I still thought of the distances between set points as being absolute, so I never thought about it in quite this way before. So what then? There is no such thing as an absolute distance a body travels between two points with a relative speed between them, not even counting relativistic effects? Are time and relative speed the only true measures of motion, then?

__________________
Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

By the way, just to be sure on this point, I am considering relative inertial speeds only, as with the catcher and thrower moving toward each other in free space, throwing spaceballs, not like the catcher is actually running toward the thrower or vice versa.

__________________
Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

I'm not sure what about this is causing you so much apparent angst: it's just Galilean relativity. In one frame, the ball travels a shorter distance at a slower speed; in another, a longer distance at a faster speed.
It's just another version of the old example of someone on a moving train throwing up a ball and catching it again. In the train's frame, the ball goes straight up and down again: the "throw" and "catch" points are identical. To an observer outside the train, the ball describes a parabola, with the "throw" and "catch" points separated by a horizontal distance equal to the velocity of the train multiplied by the time the ball is in the air.

Grant Hutchison

Yes, but there appears to be something more to it than that. Only observers in the same frame can ever agree on the distance a body has travelled, so there is no such thing as an absolute distance. There is no real way to measure a distance travelled, then, when it can vary between frames to such a degree, and the only thing all frames can agree upon to measure by is the time.

__________________
Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

What's identical in both of their reference frames is not the speed of the ball, but the speed at which the ball and the catcher converged. Thus, the original distance between ball and catcher is the same to everyone and is closed to zero in the same amoung of time to everyone; the difference is in how much of that distance was covered by the ball, and how much of it was covered by the catcher.

In the first post the poster said "forget relativity".

Motion doesn't matter anymore.

Just the ball.

It doesn't matter if both are moving or only one.

They are CLOSING THE DISTANCE.

When the ball leaves the thrower, the ditance is closing as they move, and the ball will get to the catcher when they catch it.

Nothing special or scientific without numbers (or time) involved.

Okay, now let's consider this, with relativity. Two observers are moving toward each other at a relative speed of .1c . This is the speed they observe of each other. One emits a pulse of light toward the other when they are one light-second apart. The emitter sees it moving away from him at c and the receiver sees it approaching her at c. Now, according to the frame of the receiver, the pulse travels the entire distance of one light-second at c, and so one second has passed upon her clock during the duration of the trip. But according to the emitter, the receiver was also moving toward the pulse at .1c while the pulse travelled toward her at c, so only 10/11 of a second has passed upon his clock during the duration of the trip. What am I missing here?

__________________
Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

The thing that emitted the light can't itself be moving at the speed of light.

The emitter and receiver are moving at a relative speed of .1c toward each other. Each sees the pulse of light travelling at c.

__________________
Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

Akhck, pesky decimal points...