Coming to terms with the velocity addition formula

[adimare]

I have two rockets, one in point A and another in point B, I'm going to crash one against the other, twice, and I will be observing safely from my laboratory at C, which is right in the middle between A and B.

The first time the experiment takes place, I see both rockets rushing forward to one another at a constant speed of 0.75c, they meet and crash right in the middle point between A and B.

The second time, the rocket at A is at rest relative to me, and I see the other rocket moving towards it from B at a speed of 0.96c, they meet and crash at A.

Using the velocity addition formula to calculate what the speed of B relative to A was in both experiments, we'd get precisely 0.96c, which I believe means that for the rocket at A, both experiments should produce the exact same results (let's define results as the amount of damage in the ships), which means that both experiments should produce the exact same results in every reference frame.

I'm just having trouble digesting this from the reference frame of the laboratory, where the configuration of each of the two experiments seems somewhat different as to produce the exact same results. In the first experiment, considering the laboratory frame exclusively, after one meter of time has passed, the rockets are 1.5 meters closer than they were before, whereas in the second experiment, after one meter of time has gone by, they're only 0.96 meters closer to each other.

I think I'm just trying to find a way to visualize these results without having to analyze the experiments from the reference frame of either of the ships.

 

[DaveC426913]

I'm just having trouble digesting this from the reference frame of the laboratory, where the configuration of each of the two experiments seems somewhat different as to produce the exact same results. In the first experiment, considering the laboratory frame exclusively, after one meter of time has passed, the rockets are 1.5 meters closer than they were before, whereas in the second experiment, after one meter of time has gone by, they're only 0.96 meters closer to each other.

Have you accounted for
1] time contraction when measuring simultaniety of events?
2] length contraction of the rockets?
When you do, you should find the discrepancy evaporates.

 

[Entropia]

Your observations and confusion about the velocity addition formula are completely valid. It can be difficult to wrap our minds around the idea that the same physical event can appear differently in different reference frames. However, this is a fundamental concept in special relativity and is supported by numerous experiments and observations.

The key to understanding the results of these experiments is to remember that the velocity addition formula only applies to velocities that are measured in the same reference frame. In your first experiment, both rockets are moving at 0.75c in the laboratory frame, so the velocity addition formula can be used to calculate their relative speed. However, in the second experiment, the rocket at A is at rest in the laboratory frame, so its velocity cannot be added to the velocity of the rocket at B. Instead, we must use the Lorentz transformation equations to convert the velocity of the rocket at B from the laboratory frame to the reference frame of the rocket at A. This is why we get a different result of 0.96c in the second experiment.

It may seem counterintuitive, but the results of these experiments do indeed show that the amount of damage in the ships will be the same in both reference frames. This is because the laws of physics, including the conservation of energy and momentum, hold true in all inertial frames of reference. So while the physical appearance of the events may be different, the underlying principles and final outcomes will remain the same.

In conclusion, the velocity addition formula and the concepts of relativity can be complex and challenging to understand, but they have been extensively tested and proven to accurately describe the behavior of objects moving at high speeds. It takes time and practice to become comfortable with these ideas, but keep exploring and asking questions, and you will eventually come to terms with them.