Weird paradox about relativity.?

Question

So a train is traveling near the speed of light, let's say 3/4 the speed of light (C), and a man fires a bullet on the train that travels to his perspective at 3/4 C. From someone standing outside the train the bullet should appear to travel at 1 1/2 C, but that would be impossible (Einstein). So the answer must be that each object (the bullet and the train) must travel at a combined speed of less than the speed of light. So each objects speed is limited to just less than 1/2 C. But, a ant (stretch your imagination here ) is traveling along the bullet at just less than 1/2 C. Giving him a speed of more than C which is impossible, so each objects speed must be even slower, around 1/3 C. But a gnat is traveling across the ants back...etc. If you combine the speed of MANY objects traveling within each other, you should eventually come up with a train that can't travel more than a mere crawl. Where is the flaw with this reasoning? This paradox seems to have a Zeno quality to it.

Answer 1

It's not so much a paradox about combining the speed, as it is a question of what happens as the speed of light is approached. Time will stand still, and eventually start going backward as the speed of light is approached, so the moment at which the item would have exceeded the speed of light can never be reached.



It is difficult to think of how these things work, because we are accustomed to thinking of our realitiy as a three-dimensional universe, but there is more to it than that. What we think of as straight lines in our tiny view of reality are curved as you step back and take a broader view, just as the flatness of our world's surface is revealed to be ball shaped as you move away from the Earth.



Try reading "A Brief History Of Time" by Stephen Hawking or "Einstein for Dummies" to get a handle on this stuff. I can tell by your intelligent question that you are smart enough and curious enough to learn about this and enjoy it. It's your reality!

Answer 2

flaw?....psh yea theres lots of flaws, first things first, time slows down as an object nears the speed of light (to its perspective), but that doesnt apply here now does it

now second...if an object travels faster than the speed of light, then its mass increases indefinitely (Einstein) until it get infinitely large, at which case the bullet (if possible to surpass the speed of light) wont even have enough time and distance to fit through the barrel before backfiring on the person shooting it, the ants and things on its back would be instantly crushed between the barrel and the bullet.

now third...if an ant was really on the bullet when it was fired, lets just say IT IS traveling at 3/4 the speed of light, a bullet travels at around 1/6 the speed of light, the g forces, including the inertia and resistance force of the air, the ant won't even survive through the barrel, it would be instantly crushed under the speed and air (that is imagining that the bullet doesnt increase in mass which crushes it anyways)



so yea....i think its pretty impossible for all of this to happen successfully, just....impossible...yet =D

Answer 3

You can't combine relativities. The relative speeds of the objects/insects would not approach anywhere near 1/2C or 1/3C or anything like that. You would get 0.0001C, then half of that, then half of that, and so on. The only way you would get a train that can't travel more than a mere crawl would be if the train, the bullet, the ant, and the gnat all had masses relatively the same.

Answer 4

You are using Galilean relativity where you should not be.

At relativistic speeds (speeds on the order of the speed of light), you cannot add speeds in the way you did (or tried to, but stopped yourself).



At low speeds (for example, driving down the high way), if you are traveling at 60 mph with respect to the road and you look out and see something traveling faster than you by 60 mph, you can add those speeds together to find that the other object is traveling at 120 mph with respect to the road. Doing it like this is very intuitive, and works out to be a very good approximation of the truth.

But at relativistic speeds you cannot do this. if something is traveling at 3/4 c with respect to the ground and it sees something moving at 3/4 c with respect to it...then with respect to the ground, the object is NOT moving at 1.5 c...since that would be impossible.

No massive object can exceed the speed of light in any frame of reference.



When combining relativistic speeds, you need to do something special.

This is called a Lorentz velocity transformation (look it up for a more complete explanation…it is beyond the scope of this answer right now)

v_x' = (v_x - u) / (1 - u * v_x / c^2)



It takes a bit of explaining to understand how all works, but you can use this formula (re-arranging it if necessary) to find the relative speeds of objects in multiple reference frames.



You can see that when the speeds v_x and u are small, then the denominator of the fraction becomes very closer to one and the numerator essentially equals the difference in speeds as we would expect using Galilean relativity. But as v_x and u becomes large, the denominator influences the final value more and more. If v_x * u = c^2, then everything blows up and you get an infinite answer...but this cannot happen.

Answer 5

Your flaw is your mathematics and your understanding of the laws of relativitiy...



Let's slow everything down to a realistic basis for a moment. If a train were traveling at 35 mph, and you were (some how able to be) standing atop of the train, and you fire a bullet....



Lets stop there...

We have to have a fundamental speed at which said bullet were able to travel from the barrell into open air. Each gun, bullet, and caliber each will have its own velocity upon being expulsed from the barrell, so we will give our gun's bullet a speed of... eh.. a hundred miles an hour.



back to atop of the train, hurtling through the contry side at 35 mph. If I were to fire a bullet towards the front of the train, in the direction of travel, the bullet would be traveling at 100 mph. if the train were to be traveling at 55 mph, the bullet would be traveling at 100 mph... the RELATIVITY here, is when YOU view the bullet... (if it were possible to do so) you would see the bullet travelling MUCH SLOWER than 100 mph. This is due to the fact that you are traveling with the bullet, albeit at a slower pace...



Make it easier to understand... If you are atop of a train traveling north at 99 mph, and you shoot a gun in the direction the train is travelling, the bullet will leave the gun at the speed of 100 mph... however, due to the fact you are traveling with the bullet, one mile per hour less than the bullet, the bullet will leave the gun at a much slower pace, and you should in fact be able to watch the bullet, slowly pull away from you, one mile per hour. Every hour, the bullet will be one mile more away from your location.



Here's another consequence of relativity...



The earth is constantly spinning. Yet you are sitting at your computer reading this message. As the earth rotates, it is also in orbit around the sun.



Your movement if I'm:



STANDING NEXT TO YOU: 0 miles per hour



FLOATING JUST ABOVE THE EQUATOR

OF THE EARTH WATCHING YOU: 1000 miles per hour



STANDING ON THE SUN WATCHING YOUR ORBIT ON EARTH: 67,000 miles per hour



it's all relative

Answer 6

Einstein's mathematics provide a model for addition of velocities; I don't remember the exact formula but any textbook on relativistic mechanics will have it. As seen by a "stationary" observer, the bullet fired from the train will be traveling at about 90% of the speed of light. In short, you can add any two velocites, and the sum will be less than the speed of light.

Answer 7

Speeds don't add as you would expect. Even for two velocities well below the speed of light, like v=c/10, the sum would not be c/5. You have to consider the length contractions and time dilations. While in one reference frame, the object has moved x meters in y seconds, that same object moved a different distance in a different time in the other frame. It works out so that no "combined" velocity exceeds the speed of light.

Answer 8

And God disappeared in a puff of logic. lol

Answer 9

errr.. don't bend your light saber over it.