Principle of relativity and the train and ball example I have a question about the theory of relativity. I took a lesson on brilliant.org, maybe you have heard about this educational platform.
And the example with the ball and the train is completely incomprehensible to me.
Their theory states that if a train that is moving at a speed of 200 km/h collides with a stationary ball, then the ball will move at a speed of 400 km/h from the point of view of the observer.
But I still don't understand why.
It seems to me that the ball would move with such a speed if it also moved at a speed of 200 km / h and then, upon collision, it began to move at a speed of 400 km / h.
I apologize in advance for the stupid question.
link to lesson - https://brilliant.org/courses/puzzle-science/introduction-64/science-rules-3-diagrammar/1/
 A: As others have noted, this could be called "classical" or "Galilean" relativity, which is not what the Einstein theory is that most people associate with the word.
Assuming this is an ordinary elastic collision obeying Newtonian mechanics, the ball moves with twice the speed after the train impacts it for the same reason that a falling ball bounces off the ground and returns upward. If there is no energy loss, it bounces upwards with the same speed it impacted the ground, which is a change in velocity of 2 times the initial velocity (velocity is speed with a direction). So, if you go into a frame where the ball is stationary and the ground is moving toward the ball instead, after the impact the ball is moving away with twice the speed of the ground (draw some pictures and convince yourself that is true). In the train example, the train is the earth and the velocity is horizontal rather than vertical, but the principle is the same. A massive object hitting a stationary tiny object elastically causes the tiny object to move with twice the speed of the massive one.
The ability to change reference frames, between the one where the train is moving and the ball stationary and the one where the ball is moving and the train is stationary, is basically "the principle of relativity." However, this version of it is weak, it does not actually preserve all the laws of physics.
A: According to the principle of (classical) relativity, the laws of mechanics are valid for all inertial frames of reference.
The train is moving with constant velocity, so it is an inertial frame. From the train's frame, there is a ball hitting it at 200 km/h. Supposing an elastic collision, the ball will bounce back with the same speed. The recoil of the train is supposed negligible due to its huge mass.
For a ground observer, a ball with a speed 200 km/h bigger than the train has 400 km/h.
A: Evidently your model involves an idealized perfectly elastic collision.  I am all in favor of idealized conditions.
Let’s go to the pool table.  The cue ball slams into the 8-ball, perfectly elastic collision.  The cue ball comes to a complete halt, and the 8-ball has takeaway speed EXACTLY equal to original cue ball speed.
Now let a train slam into the 8-ball.  The train is a million times more massive than the 8-ball.  The train hardly slows down at all and the 8-ball has bounce speed just about double train speed.  But not EXACTLY double, even under ideal conditions.
This post has nothing to do with Special Relativity UNLESS you consider speed v = 400 kilometers per hour in relation to speed-of-light c = 1 billion kilometers per hour.  And your problem is v²/c².  It’s maybe one part in 10¹³ .
