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Ever since my high school physics days I found relativity fascinating but I don't think I have great insight even into special relativity.

For example, in almost every lesson or video they give an example of the path of a ball thrown in a train before they introduce SR. That is, an observer who throws a ball straight up in a train will see it follow a straight path (just up and down), whereas a stationary observer will see the ball follow a parabolic path. Both observers would agree on the time it took for the ball to go up and down, but they would disagree on the length of the path the ball followed and the horizontal velocity of the ball. I don't understand why this is so significant to SR and how it relates to time dilation, length contraction, and mass increase at all.

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Since the train has no relativistic velocity, you will not see any effect such as time dilation, length contraction, lack simultaneity and so on.

The importance of this example as a prelude of Special Relativity is that it shows that even Galilean Relativity has some physical quantities which are not absolute. Namely (in this example) the length of the path covered by the ball.

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I think the purpose of that example is to show how observers from different frames of reference can disagree on what they actually observe, or measure. The example you provided can be solved with a simple galilean transformation. This example is crucial because it shows you how different observers can disagree on the same event, which seems pretty intuitive in the normal world.

However, as you start to move at relativistic speeds, certain effects such as time dilation and length contraction start to happen and without the earlier example, it can be very confusing as to why 2 different observers might disagree on the same event

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The point is exactly that agreeing on a particular value for one measured quantity causes other quantities to have different measured values, for the values in question of distance, time and velocity (any one of which can be calculated from the other two).

The limitation we're stuck with is, at its root, that we have no way to measure time passively. While you can lay a ruler next to am object and measure its length without the ruler or object doing anything, there's no equivalent for measuring time. To measure time, you have to use a clock that does something cyclically and that's where we hit trouble with classical mechanics at relativistic speeds. Keep that in mind as I expound:

In the classical ball-on-a-train example, the observer on the ground and the observer on the train agree on times and distances (relatively): their clocks measure the same seconds and in $t$ seconds they move a defined distance $x$ relative to each other. What they disagree on is velocities: each one considers himself stationary and the other moving; and when the ball is tossed, they disagree on both its speed and its direction, which comprise its velocity.

But what if it were the other way around, and the two observers agreed on the speed of the ball in all cases? So instead of arbitrarily defining the distances and measuring the elapsed time to calculate the speed, we arbitrarily define the speed, measure either the time or distance and calculate the third. We'll define the speed at 1 m/s for this examination.

Let's say one observer is on the train as it is passing a station platform on which the other observer stands. The observer on the train tosses the ball as he passes one end of the platform and catches it again as he passes the other end. In his frame of reference, he tosses the ball half a a meter up and it falls half a meter down, which at 1 m/s takes one second, so the train observer calculates that the platform is one meter long... Rather small train station, really.

Or we can view from the frame of the observer on the platform, where he measures the length of the platform with a meter stick and finds it to be fifty meters. As the train passes and the ball is tossed, he sees the ball travel about fifty meters at 1 m/s - he calculates then that the ball is in the air for about fifty seconds, making a long slow arc. From this perspective, the ball is moving in slow motion.

Note that since one is stationary with respect to the platform and thus can only measure distance, and the other has a 'clock' but no ruler, each one has to use the speed of the ball (a ratio of distance to time) as a measuring device to calculate their unknowns, and thus that value must be a constant. And if you're treating the velocity as a constant, then the distance and time measurements may be variable - and in fact will vary from one relatively moving frame to another.

Using the speed of a ball as a universal constant may sound unreasonable, but it's not far from the truth: in reality we observe the speed of light to be constant across all inertial reference frames. We can use that fact to build a clock, by bouncing light off a mirror a known distance away and counting the round trip as "1 time period," similar to throwing a ball a certain height and counting that round trip as a time period, as we did above. But our light clock has the same features as tossing a ball: treating speed as a constant forces time and/or distance to be variable. Passing objects appear flattened, passing clocks seem to run slow. This is a direct and unavoidable consequence of treating an object's speed as a fixed value across relatively moving reference frames. It really makes no difference what the object with fixed speed is... The only reason the speed of light is the relevant value in special relativity is that that speed is measured to be constant, as opposed to the purely-by-definition-constant speed of the tossed ball in our earlier thought experiment.

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