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Two spaceships in opposite direction at near c

  1. Nov 26, 2012 #1
    if you have 2 spaceships and they depart in exactly the opposite direction from a starting point, say a space station, and they accelerate to speeds nearing the speed of light, then what is the relative speed of each spaceship, using one of them as a reference frame.

    it would seem to me, that the space station would be moving at a speed nearing c, and then the other spaceship would need to be travelling at double that.

    but that can't be right.
  2. jcsd
  3. Nov 26, 2012 #2


    Staff: Mentor

    Yes, it can't be right, and it isn't. The correct formula for adding velocities in SR is:

    [tex]w' = \frac{v + w}{1 + vw / c^2}[/tex]

    where w is some object's velocity in one frame, and w' is the same object's velocity in a second frame moving at v relative to the first.

    In your scenario, say spaceship A moves off to the left from the space station, with speed w, and spaceship B moves off to the right with speed v. In spaceship B's frame, the velocity of spaceship A is then w', as given by the above formula. It should be evident that if w < c and v < c, then w' < c also.

    Picking some concrete numbers for an example, if v = .99c (to the right) and w = .99c (to the left), then

    [tex]w' = \frac{.99 + .99}{1 + .99 * .99} c = .99995 c[/tex]

    So spaceship A will be moving at .99995c relative to spaceship B.
  4. Nov 26, 2012 #3
    i see. thx, but by what mechanism does this relationship exist? meaning, what is it in the real world is this formula explaining, or, where does it come from?
  5. Nov 26, 2012 #4
    Special Theory of Relativity.
  6. Nov 26, 2012 #5


    Staff: Mentor

    The formula I gave is the relativistic velocity addition law, which is a consequence of the Lorentz transformation formulas. See the Usenet Physics FAQ for a good (if brief) discussion:

  7. Nov 26, 2012 #6
    stu dent, one way to get a physical feel for what's going on is to develop an understanding of your problem in the context of a 4-dimensional universe. I'll give you a quick picture and a link for further details just in the event you might be interested in this approach. I and others here can provide more information on this if you wish. One key element of the picture below is a strange and mysterious aspect of nature resulting in different cross-section views of the universe for observers moving at different speeds. This, in large measure, accounts for the strange things going on with special relativity theory and why velocities do not add the way you might have thought. The slant of the X4 axes for the red and blue rockets relate to speeds--but we have red, blue and black frames of reference here, and the X1 axes (worlds the different observers "live in") cut across the universe at different angles as well.

    You might begin by looking at the picture illustrating the different times for the red and blue observers (time dilation) as shown below. When the blue guy is at position 9 along his X4 axis (time axis) his world includes the red rocket located at red's frame position 8. But when red is at his position 9, the blue rocket is at the blue position 7. Keep in mind each is viewing 3-D cross-sections of 4-dimensional rockets.

    Here is a link for more details (see post #19):

    Last edited: Nov 27, 2012
  8. Nov 27, 2012 #7
    If you were able to follow the space-time diagrams in the previous post and link, then you can easily see in the sketch below how it is that the velocities add. In the previous post we had red and blue frames moving in opposite directions at the same speed with respect to the black frame. Now we look at the speeds of the red and black frames with respect to the blue frame. The black frame moves at 1/2 the speed of light with respect to the blue frame (moving in blue's negative X1 direction). The red frame is moving at 1/2 the speed of light with respect to the black frame (moving in black's negative X1 direction).

    But, now you see clearly that (1/2)c + (1/2)c does not give you 1 x c for the red frame with respect to the blue frame. Red is moving at relativistic speed in the blue negative X1 direction, but certainly not at the speed of light.
    Last edited: Nov 27, 2012
  9. Nov 27, 2012 #8
    In addition: SR does not give a "mechanism", as it is based on logical (mathematical) deduction of phenomenological principles. Historically there have been different explanations of "what is really going on". If Bob's 4D reality makes sense to you, then that is nice; alternatively you could go for Lorentz's "ether" concept or Harvey's "physical relativity" or ....
  10. Nov 29, 2012 #9
    no offense, but you may as well have said: because i said so. Which will never be satisfactory for me, even if it is einstein that said so.

    i mean, not that i doubt einstein is correct, but i want to know why he must be correct. why it is necessary for it to be that way.

    i mean, i realise it must be many steps of logic building upon each other in order to arrive at this conclusion, but there are steps that provide certainty and conclude this fact.

    if i am in a space ship, and i travel in one direction at nearly the speed of light, and another does so in the opposite direction, then i don't see how i influence that other space ship. i don't understand how my velocity, can hold implications for the velocity of another object. it makes perfect sense to me that an energy with mass cannot reach the speed of light. i get that.

    but i don't fully get why or how it can be that my velocity relative to another object that has mass can't reach or exceed the speed of light.

    what i'm thinkign now, that i will ponder further also, is that it nearly seems as though the quantity of energy required to accelerate an object would then be different depending on your reference frame.

    or, since this seems not to make a lick of sense, the relative velocity of (let's call it space ship A and B and then Earth) of A relative to earth, is the equivalent to A relative to B.

    or no, wait, A relative to light in Earth frame, is equivalent to A relative to light in B frame, which may or may not be different, which i also need to think about, and also, a curious thing would seem to me, that the closer that B approaches the speed of light then, regardless of the velocity of A, the more similar A's velocity is to earth's in B's reference frame.

    which intuitively feels to me, like the opposite of time dilation effect, because it is like putting the universe on pause, while you continue to age.
  11. Nov 29, 2012 #10
    thanks for all the replies, i will explore these links and posts, but may not get to reply until later.
  12. Nov 29, 2012 #11


    Staff: Mentor

    It doesn't. That's not the issue. The issue is, how do you *measure* the velocity of an object that is spatially distant from you? You can't observe it directly, so you have to somehow translate the direct observations you can make into a value for the distant object's velocity. Any such translation is not just dependent on your motion, or on the motion of the distant object; it's also dependent on the properties of spacetime itself, since the information about the distant object's motion has to travel to you through spacetime. And it's the properties of spacetime that give rise to the relativistic velocity addition formula.
  13. Nov 29, 2012 #12
    i think maybe mechanism was not the best word, what i was trying to convey was a hard thing to say. i meant like, why it must be.

    math is all very nice. but math is not enough. it is not a proper explanation.

    for example, we could plot 4 dimensional equations in a 2d altitude map kind of way, but for 4d, and we'd get some confusing drawings. the understanding would be lost on us. we could do operations as well, and mathematical deduce the results of these things, and try to draw more kind of maps again as well.

    but still, we would know nothing.

    but call the 4th dimension time, and suddenly everything changes. the realisation, which is not mathematical, makes everything apparent. it allows it to make sense. it provides a real world understanding. we were plotting a moving object in time frames.

    i was getting at something like that. i want to understand how it can be possible, and why it must be possible, and what exactly is happening.

    the formula should be telling for this, but i am uncertain how it got discovered, why it must be that way.

    the formula is precise directions of how. it does not show why, nor does it show why it is certain that is how.
  14. Nov 29, 2012 #13
    how did lorentz figure out lorentz transformations, and why they were necessary?
  15. Nov 30, 2012 #14
    The lorentz transformation can actually be derived by simply starting with the assumption that the speed of light in vacuum is the same for all observers (which, in fact, is an observed and measured phenomenon, rather than being a mere hypothesis). It isn't even all that complicated to understand how it's derived from there.

    The factor that Lorentz came up with was simply a description of how relative lengths contract in different frames of reference (and, as said, is derived directly from assuming that c is the same in all frames of reference.) It described the results of the Michelson-Morley experiment as well as a bunch of others.

    (This was technically speaking a hypothesis at first, because there was only a limited amount of measurement data available. However, with time it has been proven quite accurate in quite many different situations.)
  16. Nov 30, 2012 #15
    He was developing an electrodynamics theory (electrons etc) which modeled electrodynamics relative to a stationary medium, the "ether". Following the "Galilean" transformations of classical mechanics, it should be possible in theory to detect motion relative to the ether - but such experiments failed.

    In earlier discussions I gave a much simplified sketch of how he figured it out:
    https://www.physicsforums.com/showthread.php?p=3756233#post3756233 (very roughly the same approach as Einstein)
    https://www.physicsforums.com/showthread.php?p=3887942#post3887942 [/url] (simplified and incomplete derivation; and see the correction in the there following post).
    Last edited: Nov 30, 2012
  17. Nov 30, 2012 #16
    Student, don't give up on seeking the answer to that question - that question shows great insight.
  18. Nov 30, 2012 #17
    A wonderful and refreshing response, robinpike.
  19. Dec 6, 2012 #18
    Here is an example of the issue:

    Two space ships accelerate in opposite directions from a space station in deep space. The space station calculates the speed of each departing space ship relative to itself by sending out light signals to the space ships, which they return straight back to the space station.

    In this way, the people on the space station can calculate the speed of each space ship relative to themselves. Eventually, the two space ships reach individual departing speeds from the space station of 3/4 the speed of light.

    This suggests that the people on the space station see the two space ships recede away from each other at 1.5 times the speed of light?

    And yet the two space ships can still communicate with each other by sending light signals to each other.
  20. Dec 6, 2012 #19
    That is not an issue (and different from the OP), except if it is still an issue for you if your replace "light in space" by "sound in air". There is in principle no problem for two airplanes to exchange sound signals if they recede away in opposite directions at 3/4 the speed of sound.
  21. Dec 6, 2012 #20


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    Science Advisor
    Gold Member

    Yes, that's because, as PeterDonis pointed out in post #2, they see each other receding away from each other at:

    w' = (.75 + .75)/(1 + .75 * .75) c = .96 c
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