# Relative Velocity of two particles

1. May 5, 2010

### Starwanderer1

Dont know if this has been discussed here before..
If two relativistic particles are travelling with speeds 'u' &'v' ,how to calculate the relative velocity?

2. May 5, 2010

3. May 5, 2010

### Starwanderer1

Much thanks!!!

4. May 5, 2010

### starthaus

Depends:

1. If the particles are travelling at speeds "u" respectively "v" wrt a common frame of reference then the answer is u+v if they travel towards each other or u-v if they are travel in the same direction.

2. If particle A travels at speed "u" wrt a frame of reference and particle B travels at speed "v" wrt A then the answer is simply v.
(the speed "u" has no influence on the answer, this is not a problem about relativistic speed composition)

Judging by the way you asked your question, the answer is (surprisingly) the one at point 1.

Last edited: May 5, 2010
5. May 5, 2010

### Fredrik

Staff Emeritus
Starthaus, you got most (if not all) of that wrong. Would you like to try again?

6. May 5, 2010

### starthaus

Yes, the second half of the answer was wrong, I wasn't paying attention, I corrected it.
The fact stands that the answer is option 1 and that, in any case, the answer has nothing to do with relativistic speed composition. In fact, the correct answer is orthogonal to relativity altogether.
Apparently the notion of "closing"and "separation" speed are not well known. The answer to the Op's question is, suprisingly, option 1. It is very simple to prove to yourself:

If two particles travel towards each other at speeds u and v measured in the SAME frame, then, they cover the distance between them according to the equation:

$$X=x_1+x_2$$

So:

$$w=\frac{dX}{dt}=\frac{dx_1}{dt}+ \frac{dx_2}{dt}=u+v$$

If the particles "chase" each other, then, their "separation" is:

$$X=x_1-x_2$$

so, their relative speed is:

$$w=\frac{dX}{dt}=\frac{dx_1}{dt}- \frac{dx_2}{dt}=u-v$$

The above rules work just the same for accelerations and all the higher time derivatives.
Closing and separation speed work according to very different rules than the speed composition in SR. Neither answer (option 1 or 2 ) has anything to do with the relativistic composition of speeds.

Last edited: May 5, 2010
7. May 5, 2010

### Ich

Guess what, according to Wikipedia, as well as the links therein, starthaus is right. There seems to be some kind of edit war, but I'd be interested whether there are reliable sources for this, IMHO nutty, definition. Can it be that there are serious relativists who define it as the vector difference of velocities in a common IS? Any sources?

8. May 5, 2010

### matheinste

Closing or separation speed has often come up and been explained in this forum although I cannot immediately give a link. The closing/separation speed of two material objects can approach 2c. But of course their velocity relative to each other is less than c as measured in either of their rest frames and no material object moves faster than c.

Matheinste.

9. May 5, 2010

### DrGreg

Which Wikipedia article? There are dozens about various aspects of relativity.

10. May 5, 2010

### Ich

Yes, I know there were attemtps here to give the vector difference a name that distiguishes it from relative velocity. My concern is that the WP article calls exaclty this quantity the "relative velocity". There are also links from trustworthy sites that give the same definition, albeit always in a non-relativistic context.
Now that one is really scary.

11. May 5, 2010

### starthaus

thank you, Ich

The answer has nothing to do with relativity, it is just basic physics.
Besides, it is not a definition, it falls straight out of basic math.

12. May 5, 2010

### Ich

http://en.wikipedia.org/wiki/Relative_velocity" [Broken]

Last edited by a moderator: May 4, 2017
13. May 5, 2010

### Ich

Hi starthaus

Well, as you can see, I acknowledge that there is some support for your point of view, so it's arguably inappropriate to call it wrong. That came as a surprise to me, and I bet quite a few other PF members are irritated, too.
However, I disagree with your (first) statement and with some of the definitions I've seen in the web. But this is a matter of semantics, where the only fault can be to use a nonstandard definition without explicitly saying so. I hereby grudgingly admit that you obviously did not use a nonstandard definition of the term.
Still, I'd be surprised if recent textbook authors on relativity really used that definition.
I disagree. IMHO, a resonable definition of the relative velocity of A and B would be B's velocity as measured by A, or the other way round. The definition (yes, it is a definition, not basic math) as a vector difference is theory dependent - even worse, depends on a theory known to be inaccurate for over 100 years now.

14. May 5, 2010

### matheinste

Rindler in Relativity, Special, General and Cosmological. 2nd ed. Page 70 gives an equivalent vector formula and calls it, to distiguish it from relative velocity, mutual velocity and states that it is applicable equally to Newtonian and Relativistic kinematics.

In Essential Relativity. 2nd ed. Page 36 he also calls it mutual velocity and describes it as the time rate of change of the vector connecting the two particles in question.

Matheinste.

Last edited: May 5, 2010
15. May 5, 2010

### starthaus

The article is perfectly correct. Basic stuff, something that gets taught in 9-th grade.

Last edited by a moderator: May 4, 2017
16. May 5, 2010

### starthaus

Yes, nothing exotic, basic physics. What's good for the Rindler is good for the goose :-)

Last edited: May 5, 2010
17. May 5, 2010

### starthaus

Thank you, it is nice to see that sometimes people admit to be rash in their judgement. I have not seen it very often in this forum.
There is no reason to be irritated, this is perfectly basic physics (see Rindler "Relativity, Special, General and Cosmological". 2nd ed. Page 70)

I did not use a nonstandard definition, moreover, I provided the mathematical support as to how we arrive to the expressions.

See Rindler, exact page cited by "matheinste".

I think you share the same confusion with the other science advisors (jtbell and Fredrik-who, BTW, reacted quite violently but without any reason) , closing speed does NOT contradict relativity.
The closing/separation speed is the answer to the question posed in the OP. Relativistic speed composition is not the correct answer and, in fact, has nothing to do with the problem posed by the OP.

Last edited: May 5, 2010
18. May 5, 2010

### Staff: Mentor

Give us a break! Those advisors are well aware of the definition of closing speed and how it differs from the usual definition of relative velocity.
Really? The OP asked about the relative velocity, which is usually taken to mean the velocity of one object as measured in the frame of the other.

19. May 5, 2010

### starthaus

then the answer is Option 2 (i.e. $$v$$).
The point is that the correct answer has nothing to do with relativistic speed composition, so jtbell's reference to the hyperphysics page is wrong.

20. May 5, 2010

### DrGreg

The original poster specifically asked about "relative velocity". In relativity, the "relative velocity of A relative to B" is assumed to mean "as measured by B" unless it's explicitly clear that the assumption is wrong. This is (uv)/(1−uv/c2).

The "relative velocity of A relative to B as measured by C" (where C is not B) is not encountered very frequently. It is (uv), where both u and v are measured by C. To avoid confusion it is often referred to as "closing velocity" or, as Rindler says, "mutual velocity".

The Wikipedia article on Relative velocity makes no mention of relativity in the text and therefore must be assumed to be about non-relativistic Newtonian mechanics, where this issue doesn't arise. In any case, I would dispute the wording of the definition given in the opening sentence. Please remember Wikipedia can be edited by anyone who wants to and therefore cannot be relied on for authority.

The point is, "closing velocity" (in this sense) is not the same as "relative velocity"; the question was phrased in terms of "relative velocity" and it's reasonable to assume that was what was meant unless the original poster would like to tell us otherwise.

21. May 5, 2010

### Staff: Mentor

That hyperphysics reference was a perfectly reasonable response to the OP's question. Lacking further details of what the OP actually intended, assuming 'u' and 'v' to be the speeds of the particles with respect to some third frame is entirely reasonable. (And seemed to satisfy the OP.)

22. May 5, 2010

### starthaus

This is a nice, balanced answer, devoid of emotion. It underscores the importance of specifying the reference frame used for judgement in answering the question. You need to ask the OP what exactly was the reference frame for:

-measuring u and v
-measuring the relative speed between A and B

For example, if the speed (v) of particle B was specified wrt the frame co-moving with particle A then their relative speed as measured wrt A is ...v.

The naming convention varies depending on the author (see the wiki case). So, going by the naming convention is not a good way of solving the problem. Specifying the frame of reference is.

Last edited: May 5, 2010
23. May 5, 2010

### starthaus

Yes, it is reasonable, once you specify the frame of reference.
So are the other two choices.

24. May 5, 2010

### Fredrik

Staff Emeritus
I think it would be absurd to use the term "relative velocity" for the time derivative of the difference between the position coordinates of the two objects. When we say that a person walking inside a train has the velocity 5 m/s relative to the train, we mean that his velocity is 5 m/s in the inertial frame in which the train is stationary. To have "relative velocity of" mean something completely different than "velocity relative to" would only make sense if we wanted to cause as much confusion as possible.

It's too basic. That's the problem with it. No one knows relativity in 9th grade, so no need to distinguish between "my speed relative to the train is 5 m/s" and "the difference between my speed and the train's speed, both relative to the ground, is 5 m/s".

It's not physics at all. It's just semantics. We're just talking about what the term "relative velocity" means, and what it should mean.

I did, and as Matheinsteine already said, Rindler doesn't define relative velocity this way. The exact quote is:

"We call this, for lack of a better name, the mutual velocity between the particles in S, to distinguish it from the relative velocity, which is what one particle ascribes to the other."​

Do you seriously think that we believe that it does, or is this just an insult?

You are incorrectly assuming that your interpretation of "relative velocity" is correct and all others wrong.

25. May 5, 2010

### starthaus

Let me clarify, once and for all:

1. If u and v are specified wrt a frame C, then the relative speed of B wrt A as measured in frame C is v-u

2. If v is specified in frame of particle A then the relative speed of B wrt A is....v.

3. If u and v are specified wrt a frame C then the relative speed of B wrt A as measured wrt A is (u-v)/(1-uv/c^2)

The above is a description that is devoid of any naming convention, it is based solely on the choice of reference frames, as it should be.
Like I said, you need to be precise in terms of defining the frames of reference.