# Relative velocity of third object

newTonn
With reference to the sketch attached,
if A and B is having a relative velocity of 0.9c,and
B and C having a relative velocity 0.9c
What will be the relative velocity between A and C?
(all in x direction)

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pam
Just use the relativistic formula for addition of velocities. If you don't know it, look at a book.

newTonn
Just use the relativistic formula for addition of velocities. If you don't know it, look at a book.
Ok i looked at the book and got an answer of 0.9944c(approx).
How can i justify this with the relative displacement after one second?

Mentor
How can i justify this with the relative displacement after one second?
What do you mean? According to A, C moves 0.9944c*1s closer in one second.

newTonn
What do you mean? According to A, C moves 0.9944c*1s closer in one second.
Sorry i din't mean it.Initally let all of them have same x co-ordinates =0; According to A,what will be the positions of B & C in after 1 second in A's frame?

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Homework Helper
Since you now know the velocity of each of B and C relative to A just multiply by 1 second to find how far they have moved, relative to A in one second.

Mentor
Initally let all the points have same x co-ordinates =0; According to A,what will be the positions of B & C in after 1 second in A's frame?
According to A, B has a speed of -.9c. So B will be at x = -.9c*1sec (~ -2.7 10^8 m) after one second. Similarly, C will be at position x = -.9944c*1sec (~ -3 10^8 m).

newTonn
According to A, B has a speed of -.9c. So B will be at x = -.9c*1sec (~ -2.7 10^8 m) after one second. Similarly, C will be at position x = -.9944c*1sec (~ -3 10^8 m).
Is there any length contraction to be considered or not?

Mentor
Is there any length contraction to be considered or not?
Each observer will view the "metersticks" used by the others to be contracted. But you don't need to explicitly use that fact to figure out the coordinates of each at a given time, since you are not changing reference frames.

newTonn
Each observer will view the "metersticks" used by the others to be contracted. But you don't need to explicitly use that fact to figure out the coordinates of each at a given time, since you are not changing reference frames.
Ok what about if B and C are long rods with length 'c' in their frames?

Mentor
length contraction

Ok what about if B and C are long rods with length 'c' in their frames?
If B and C carry long rods (held parallel to the direction of motion) that have length $L_0$ in their own frames, then the length of those rods as measured by A will be:

$$L = L_0/\gamma$$

Where:
$$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$

And v is the relative velocity of B or C with respect to A (discussed earlier).

newTonn
If B and C carry long rods (held parallel to the direction of motion) that have length $L_0$ in their own frames, then the length of those rods as measured by A will be:

$$L = L_0/\gamma$$

Where:
$$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$

And v is the relative velocity of B or C with respect to A (discussed earlier).
This shortening will be from one end or it would be symmetrical from centre of Rod?

Mentor
This shortening will be from one end or it would be symmetrical from centre of Rod?
I don't understand what you mean. All segments of the rod will be uniformly "shortened".

Antenna Guy
I don't understand what you mean. All segments of the rod will be uniformly "shortened".

I think he's asking if the rod will also shrink in dimensions not parallel to the direction of travel.

Regards,

Bill

newTonn
I don't understand what you mean. All segments of the rod will be uniformly "shortened".
Ok.i will come to the point.Consider A,B and C are at rest to each other.B & C are each carrying a rod of length 'L'.This rods are made of 'n' number of molecules in x direction.Now let us say the spacing from centre to centre of molecules are 's'.Let us mark it on C's Rod.
Now B,with the rod goes for a trip in x direction.He came back to cross his initial position with a relative velocity of 0.9c(in x-direction).The rod now is contracted as per A.Does the spacing between the molecules of the rod in B's hand change,with respect to the spacing marked on C's rod(in A's frame).?
If so,can we say the space itself is contracted ?

Mentor
Ok.i will come to the point.Consider A,B and C are at rest to each other.B & C are each carrying a rod of length 'L'.This rods are made of 'n' number of molecules in x direction.Now let us say the spacing from centre to centre of molecules are 's'.Let us mark it on C's Rod.
Now B,with the rod goes for a trip in x direction.He came back to cross his initial position with a relative velocity of 0.9c(in x-direction).The rod now is contracted as per A.Does the spacing between the molecules of the rod in B's hand change,with respect to the spacing marked on C's rod(in A's frame).?
I don't quite understand what you're asking, especially in that last sentence. (Not sure what you mean by "spacing marked on C's rod in A's frame".)

In any case, I'll rephrase what I already said. You can imagine each rod as having markings every meter (or other interval of your choice). According to frame A, the markings on B and C are closer together by the factor given in post #11. (Similar statements can be made about measurements according to any of the three frames.)

newTonn
I don't quite understand what you're asking, especially in that last sentence. (Not sure what you mean by "spacing marked on C's rod in A's frame".)

In any case, I'll rephrase what I already said. You can imagine each rod as having markings every meter (or other interval of your choice). According to frame A, the markings on B and C are closer together by the factor given in post #11. (Similar statements can be made about measurements according to any of the three frames.)
Actualy i mean the centre to centre distance of molecules(or atoms of the rod)
I will try to explain my question with another example.A,B,C and D are four persons.Initialy they are all at rest with respect to each other.
B,Cand D are in a line in x direction.
C is in the middle of B and D
C is holding a long bar whose end are just touching B and D.
Now this B,C and D started a Journey in x direction with same velocity(they are at rest with respect to each other).
They all came back in the x direction to cross A with 0.9c velocity with respect to the stationary observer A.
Do A see the ends of the rod still touching B and D or not?

Mentor
Actualy i mean the centre to centre distance of molecules(or atoms of the rod)
I will try to explain my question with another example.A,B,C and D are four persons.Initialy they are all at rest with respect to each other.
B,Cand D are in a line in x direction.
C is in the middle of B and D
C is holding a long bar whose end are just touching B and D.
Now this B,C and D started a Journey in x direction with same velocity(they are at rest with respect to each other).
They all came back in the x direction to cross A with 0.9c velocity with respect to the stationary observer A.
Do A see the ends of the rod still touching B and D or not?
It all depends on how B, C, and D accelerate to reach their final speeds. If they do it in such a way as to maintain a constant proper length between themselves, then their distances will be uniformly Lorentz contracted as seen by A--and everyone will agree that the ends of the rod still touch B and D. (Note that this is equivalent to having A accelerate to speed 0.9c, while the others remain at rest in their original inertial frame.)

Whether the ends of the rod still touch B and D (or not) will be a fact that all observers will agree upon.

newTonn
It all depends on how B, C, and D accelerate to reach their final speeds. If they do it in such a way as to maintain a constant proper length between themselves, then their distances will be uniformly Lorentz contracted as seen by A--and everyone will agree that the ends of the rod still touch B and D. (Note that this is equivalent to having A accelerate to speed 0.9c, while the others remain at rest in their original inertial frame.)

Whether the ends of the rod still touch B and D (or not) will be a fact that all observers will agree upon.
In fact the space itself is contracted isn't it?.
If only the rod was contracted,the ends would not have touch B and D.
The contraction of space leads to an ambiguity.Because the formula for velocity(derived from time dilation) doesn't take into consideration of this reduced distance.

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Mentor
In fact the space itself is contracted isn't it?.
If only the rod was contracted,the ends would not have touch B and D.
Yes, all distances are contracted.
The contraction of space leads to an ambiguity.Because the formula for velocity(derived from time dilation) doesn't take into consideration of this reduced distance.
What ambiguity in what formula? Velocity = Distance/Time works just fine as long as you stick to a single frame.

newTonn
Yes, all distances are contracted.

What ambiguity in what formula? Velocity = Distance/Time works just fine as long as you stick to a single frame.
Ok let us stick to one stationary frame from where the observation are made.
Consider another object moving with a velocity of 0.9c (all movements are in x direction)w.r.t the frame.
For this statement to be true, the object,after 1 seconds should be at a distance of 0.9 x (3E+8) =( 2.7E+8)meter.
Since all distances are contracted,the equvalent distance of (2.7E+8) as observed from stationary frame ,after contraction will be(1.305E+8)meter -(using length contraction formula).
This means he is observing the other object moving (1.305E+8)meter in x direction in one second in his clock.
Now if we calculate the velocity with this observed distance,the velocity will be 0.435c.
Please note that,the distance and time here are as observed from the stationary frame.
This is a loop.no escape out of it.

Mentor
Ok let us stick to one stationary frame from where the observation are made.
Consider another object moving with a velocity of 0.9c (all movements are in x direction)w.r.t the frame.
For this statement to be true, the object,after 1 seconds should be at a distance of 0.9 x (3E+8) =( 2.7E+8)meter.
That's correct. The object will have moved that distance (d = vt) according to the stationary frame's measurements.
Since all distances are contracted,the equvalent distance of (2.7E+8) as observed from stationary frame ,after contraction will be(1.305E+8)meter -(using length contraction formula).
He's not moving with respect to himself, so why would he use the length contraction formula?? The purpose of the length contraction formula (and other consequences of the Lorentz transformations) is to be able to relate measurements made in one frame to corresponding measurements made in another. In this case, all measurements were made in the same frame, so no such transformations are needed.
This means he is observing the other object moving (1.305E+8)meter in x direction in one second in his clock.
Now if we calculate the velocity with this observed distance,the velocity will be 0.435c.
Please note that,the distance and time here are as observed from the stationary frame.
Incorrect.
This is a loop.no escape out of it.
You are trapped in a loop of your own making, I'm afraid.

newTonn
He's not moving with respect to himself, so why would he use the length contraction formula?? The purpose of the length contraction formula (and other consequences of the Lorentz transformations) is to be able to relate measurements made in one frame to corresponding measurements made in another. In this case, all measurements were made in the same frame, so no such transformations are needed.
Length contraction, according to Hendrik Lorentz, is the physical phenomenon of a decrease in length detected by an observer in objects that travel at any non-zero velocity relative to that observer. This contraction (more formally called Lorentz contraction or Lorentz-Fitzgerald contraction) only becomes noticeable, however, at a substantial fraction of the speed of light; and the contraction is only in the direction parallel to the direction in which the observed body is travelling.

The above is from Wikipedia.
I am confused.Realy confused.
Please explain me why the observer in rest frame should not use the length contraction formula for observing other Object.Moreover you have no disagrement that all the distance there are also contracted.

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Mentor
The above is from Wikipedia.
I am confused.Realy confused
There's nothing wrong with that quote, though I would strongly recommend getting a real textbook if you want to learn relativity. (Don't try to learn physics via Wiki!)

Think of it this way. I'm in my stationary frame which extends forever in all directions. I've already got distances mapped out and marked all over the place. In particular, I've got a marker at the x = 2.7E+8 meter position which can detect when something passes by it.

This fast moving object moves past the origin at t = 0 (according to my clock, of course) and 1 second later (again, according to me) it passes the above-mentioned marker. So I deduce that it must have been moving at 0.9c. That's what it means to have such a speed. All distances and times are measured by me, so I don't need any length contraction or time dilation formulas.

Of course, another frame--such as that of the moving object--will have a different set of measurements.

Mentor
Please explain me why the observer in rest frame should not use the length contraction formula for observing other Object.
If the stationary observer were to measure the length of the moving object, then you could predict his result using the length contraction formula. But he's not doing that here.
Moreover you have no disagrement that all the distance there are also contracted.
As measured from a moving frame, not a stationary one.

newTonn
If the stationary observer were to measure the length of the moving object, then you could predict his result using the length contraction formula. But he's not doing that here.

As measured from a moving frame, not a stationary one.
Why can't We use the same principle for measuring and predicting the velocity of the object in other frame?
For example as in the previous example,
Observed velocity = V' = 0.9c
observed distance traveled per second L'= 2.7E+8 meters.
Actual distance traveled per second $$L = L'/\gamma$$ = 6.2E+8 meters.
Predicted velocity = 6.2E+8/3.0E+8
= 2.068c.
What is limiting us to do this tranformation?

Mentor
From the moving frame...

Why can't We use the same principle for measuring and predicting the velocity of the object in other frame?
For example as in the previous example,
Observed velocity = V' = 0.9c
observed distance traveled per second L'= 2.7E+8 meters.
Actual distance traveled per second $$L = L'/\gamma$$ = 6.2E+8 meters.
Predicted velocity = 6.2E+8/3.0E+8
= 2.068c.
What is limiting us to do this tranformation?
Viewed from the moving frame of the object:

The distance traveled is $L = L_0/\gamma$ = (2.7E+8)/2.29 = 1.18E+8 m.

Since the moving observer sees the stationary clocks as out of synch. The clock at the second marker is ahead of the one at the origin by 0.81 seconds, which means that the moving observer sees that only .19 seconds have passed by according to stationary clocks. This means that according to his moving clock, a time of $$t = t_0 \gamma$$ = 0.19*2.29 = .435 seconds has passed.

Work out the speed as measured by the moving object (using his time and distance scales, of course) and you'll get, once again, the speed of 0.9c = 2.7E+8 m/s.

Mentor
2021 Award
Why can't We use the same principle for measuring and predicting the velocity of the object in other frame?
For example as in the previous example,
Observed velocity = V' = 0.9c
observed distance traveled per second L'= 2.7E+8 meters.
Actual distance traveled per second $$L = L'/\gamma$$ = 6.2E+8 meters.
Predicted velocity = 6.2E+8/3.0E+8
= 2.068c.
What is limiting us to do this tranformation?
You can certainly make that calculation but you are mixing measurements from different reference frames, so the result of the calculation is not a velocity in any frame. I believe that calculation is generally called rapidity or celerity. Rapidity is not limited to c. Rapidity values greater than c are not faster than light since the rapidity of light is infinite (undefined).

You may want to do a little research on your own and check these statements. I have never actually used rapidity, so I could easily be making an error.

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newTonn
You can certainly make that calculation but you are mixing measurements from different reference frames, so the result of the calculation is not a velocity in any frame. I believe that calculation is generally called rapidity or celerity. Rapidity is not limited to c. Rapidity values greater than c are not faster than light since the rapidity of light is infinite (undefined).

You may want to do a little research on your own and check these statements. I have never actually used rapidity, so I could easily be making an error.
Yes I agree there is something confusing in my calculation.the equation should be L= L' x gamma.But the result is as per that.
Basicaly the actual length(or distance travelled) in the moving frame will be more than that of observed from stationary frame.The actual time elapsed in the moving frame will be less than what elapsed in the stationary frame(i didn't used this actual elapsed time-By using so we will get a higher value than what i obtained).
So the actual velocity will be different from that observed from stationary frame and can attain a value higher than that of 'c'.
You believe that this calculation is rapidity or celerity.

Gold Member
Yes I agree there is something confusing in my calculation.the equation should be L= L' x gamma.But the result is as per that.
Basicaly the actual length(or distance travelled) in the moving frame will be more than that of observed from stationary frame.The actual time elapsed in the moving frame will be less than what elapsed in the stationary frame(i didn't used this actual elapsed time-By using so we will get a higher value than what i obtained).
So the actual velocity will be different from that observed from stationary frame and can attain a value higher than that of 'c'.
You believe that this calculation is rapidity or celerity.
Actually celerity (also known as proper velocity) is the other way round. In the example being discussed, the celerity of the "moving" object relative to the "stationary" frame is distance measured in the "stationary" frame divided by time measured in the "moving" frame.

Celerity is larger than velocity, because time measured in the "moving" frame is shorter than time measured in the "stationary" frame. And the celerity of light is infinite. At low speeds where relativistic effects are negligible, celerity is almost identical to velocity.

Celerity = $$\gamma$$ x velocity​

Although some people have described celerity as rapidity, "rapidity" usually means something else ("hyperbolic angle").

Mentor
2021 Award
Actually celerity (also known as proper velocity) is the other way round. In the example being discussed, the celerity of the "moving" object relative to the "stationary" frame is distance measured in the "stationary" frame divided by time measured in the "moving" frame.

Celerity is larger than velocity, because time measured in the "moving" frame is shorter than time measured in the "stationary" frame. And the celerity of light is infinite. At low speeds where relativistic effects are negligible, celerity is almost identical to velocity.

Celerity = $$\gamma$$ x velocity​

Although some people have described celerity as rapidity, "rapidity" usually means something else ("hyperbolic angle").
Thanks for the clarification. Having never used it I was concerned that I may have been making exactly that mistake.

Mentor
2021 Award
So the actual velocity will be different from that observed from stationary frame and can attain a value higher than that of 'c'. You believe that this calculation is rapidity or celerity.