# Equal and opposite reaction at light speed

My knowledge of physics is very limited. And this question is only really to help me put to rest a question i have had for some time.

I would like the answer to be a simple one as i am not really in need of any complex mathematical equation or anything like that. (however the answer must be comprehensive enough to convince me its true)

So first things first.

If every action has an equal and opposite reaction then:

If i was in space and throw a ball we would both move away from each other at the same speed!. I am quite sure that's correct. (Both the ball and myself have the same mass)

The problem comes when i then throw a second ball. I accelerate some more and the ball moves away at the same speed as the first ball i through. (assuming i am always throwing balls in the same way)

So assuming this is correct. And assuming i have an infinite number of balls i can keep accelerating my self.

So once i get to the speed of light. If i throw another ball why do i not accelerate faster then the speed of light?

Assuming for the sake of argument that i have not burnt up etc

If i don't go past the speed of light because the rules don't allow it, then what happens to the ball i just through? Is it at the same speed as the first ball i through (relatively) and if so then what append to the equal and opposite force?

Or is it that i simply cannot throw the ball at all?

I am sure you all know the answer to this (: its sounds like 101 speed of light stuff!

• "If I was in space and throw a ball we would both move away from each other at the same speed. I am quite sure that's correct." Except in the case of an extraordinary coincidence (namely that you and the ball have equal masses), I am quite sure that it's not. (Unless you mean that your speed relative to the ball is equal to the ball's speed relative to you, which is obviously true, but not, I think, what you mean.) – WillO Oct 1 '15 at 0:15
• In addition to the previous commenter, as your speed goes up and gets closer to $c$, your momentum starts getting relativistic, in a sense (but not entirely correct) you'd be gaining mass, making further acceleration harder and harder: hyperphysics.phy-astr.gsu.edu/hbase/relativ/relmom.html – Gert Oct 1 '15 at 0:20
• How do you get the 2nd and other balls if you have the same mass as the first ball, and presumably the 2nd and 3rd? If you have an infinite number of balls, you have infinite mass, right? This question falls apart at the beginning and doesn't make any sense. – Bill N Oct 1 '15 at 2:27
• @BillN: I upvoted your comment, but maybe we can rescue the question by assuming that you start with one ball, you throw the ball, then a second ball drops vertically from the sky into your hand, you throw that ball, then a third ball drops vertically from the sky into your hand, you throw that ball, etc. – WillO Oct 1 '15 at 5:39

To an extent, you can exceed the speed of light exactly as you described. Because of time dilation and the Lorentz contraction, if you measure an object at 100 light years away, then you accelerate to some speed really close to light speed, it will take you less than 100 perceived years to get there. While you're traveling, the object will seem to be closer than 100 light years, and time dilation will ensure that light still appears to be traveling the same relative speed.

Of course, the rest of the universe will say it took you a bit over 100 years to get there at a bit less than the speed of light, so even if you apparently go 100 times the speed of light each way (and perceive 2 years passing for the round trip), you'll come back to meet your grandkids' grandkids.

As far as your speed relative to everyone else, it's an issue of momentum. Every time you accelerate, you add some kinetic energy to your body. The universe considers kinetic energy to be part of your total mass. At very low speeds, the mass equivalent of, for example, 100 mph of kinetic energy is tiny, so you don't notice it. But at high speeds, that effective mass starts to really matter.

So you throw a baseball and it transfers some momentum to you. But that momentum is a form of kinetic energy, which means adding momentum adds effective mass. Because you have more mass each time you throw a ball, the same momentum has a smaller effect on your velocity. The particular relationship our universe uses for mass/energy equivalence means mass approaches infinity as your speed approaches light speed. In turn, the change in velocity approaches zero near the speed of light.

Even if you magically were at exactly the speed of light, your effective mass would be infinite, so any finite momentum you add would have zero impact on your velocity.

• It was difficult yo pick the best answer here as all 3 clearly have details that deserve consideration. However this answer is the type of clean answer i was hoping for where someone who knows what they are talking about is able to see past the inaccuracies of my question and still offer an answer that i am happy with. – Paul Spain Oct 1 '15 at 18:57

The answer to your question depends on a lot of information you haven't given us.

First, as Bill N. points out in a comment, if you start out with infinitely many balls, then you've got an infinite amount of mass, so throwing a ball will not cause you to recoil. So the first piece of missing information is: Where do you keep getting new balls?

I'm willing to assume that balls keep dropping out of the sky and landing in your hands as you need them (and that these are not pushing you downward because the ground is holding you up).

But next, and far more importantly, how frequently are you throwing these balls, and how fast? Are you throwing one ball every minute according to earth clocks or one ball every minute according to your own wristwatch? Those are very different things, because people in motion with respect to each other measure time differently. (In fact, each time you throw another ball and accelerate, you'll revise your own opinion about the time intervals between the balls you've already thrown).

And next, how fast are you throwing the second (or third or fourth) ball compared to the first? Are you throwing them all at the same speed as measured by an observer on earth? Or all at the same speed as measured by you? Those are very different things, because you and the earth observer will disagree both about time measurements and about distance measurements. (And once again, you will disagree with your own past self about these things!).

Once you specify the answers to all those questions, you'll have a well defined problem. The solution to that problem will depend on your answers. But in no case will the solution be that you can accelerate to the speed of light.

• In addition, each ball would have to have zero forward velocity relative to each of the new inertial reference frames into which you jump when you throw a ball. The conclusion is the same: you never reach the speed of light, because in every new frame, a light signal will move away from you at $c$. – Bill N Oct 1 '15 at 15:18

So once i get to the speed of light. If i throw another ball why do i not accelerate faster then the speed of light?

But there's the rub: you may get arbitrarily close to c, but you'll never reach it.

As to why this occurs, I'm afraid you'll need to actually study some special relativity. One way of looking at it is that, as you go faster and faster an observer who is at what you call "rest", or zero velocity, will observe a) your length along the direction of flight will become shorter, and b) your onboard clock will run slower. So a ball thrown out the back with what you think is constant velocity will appear to be moving more slowly relative to your ship, and will accelerate you less and less as you approach c. But please, this answer is wildly oversimplified in deference to your apparent lack of the necessary math.

Relativity without the necessary math really makes no sense. The behavior of relativistic (significantly near c - and that depends on your precision of measurement) velocity is so entirely different from everyday experience that trying to explain it in common-sense terms is pretty hopeless.