B How can I derive the law of composition of velocities?

[AhmedHesham]

How can
I derive mathematically law of composition of
velocities from time dilation and length
contraction
But please use only algebra .Don't use four vectors
and space time diagrams.

 

[Ibix]

You can't. You need the relativity of simultaneity too.

Assuming you meant that, or meant the Lorentz transforms rather than time dilation and length contraction, thirty seconds with Google will find an answer - even the Wikipedia page has the derivation.

 

[SiennaTheGr8]

I don't think you can do that from time dilation and length contraction directly. You'll need the Lorentz transformation. Is that in your toolkit yet? You might find this PDF by David Morin helpful:

www.people.fas.harvard.edu/~djmorin/chap11.pdf

He builds up to the Lorentz transformation from time dilation and length contraction (and the relativity of simultaneity), and then uses the Lorentz transformation to derive the velocity-composition formula.

 

[AhmedHesham]

I don't think you can do that from time dilation and length contraction directly. You'll need the Lorentz transformation. Is that in your toolkit yet? You might find this PDF by David Morin helpful:

www.people.fas.harvard.edu/~djmorin/chap11.pdf

He builds up to the Lorentz transformation from time dilation and length contraction (and the relativity of simultaneity), and then uses the Lorentz transformation to derive the velocity-composition formula.

Thank you

 

[AhmedHesham]

You can't. You need the relativity of simultaneity too.

Assuming you meant that, or meant the Lorentz transforms rather than time dilation and length contraction, thirty seconds with Google will find an answer - even the Wikipedia page has the derivation.

Thanks for answering the question. What do you mean ?is not time dilation and length contraction enough? Why?

 

[Nugatory]

Thanks for answering the question. What do you mean ?is not time dilation and length contraction enough? Why?

Time dilation, length contraction, and relativity of simultaneity will all be present if anyone of them is. Thus, you always have to allow for all three.

 

[Sorcerer]

Thanks for answering the question. What do you mean ?is not time dilation and length contraction enough? Why?

Take this comment with a grian of salt, as I have not even gotten my BS yet, but to get the velocity transformation you have to divide the entire distance transformation by the entire time transformation. However, length contraction and time dilation are only special cases of using the Lorentz transformation-where Δt and Δx’ are zero, respectfully (because when measuring a length, you look at the ends simultaneously when doing your measurement, hence Δt = 0, and when measuring a time interval, the observer is right next to his/her clock and unmoving with respect to the event for which the time interval is measured, hence Δx’ = 0).

I.e., you can get length contraction by starting with Δx’ = γ(Δx + vΔt) and letting Δt = 0, and you can get time dilation by starting with cΔt = γ(cΔt’ + [v/c]Δx’) and letting Δx’ = 0.

That would be insufficient to represent all cases, as distance or time are not always zero.

 

[vanhees71]

How can
I derive mathematically law of composition of
velocities from time dilation and length
contraction
But please use only algebra .Don't use four vectors
and space time diagrams.

This is like asking an actor to perform Shakespeare without using the adequate language (English ;-)).

 

[Ibix]

This is like asking an actor to perform Shakespeare without using the adequate language (English ;-)).

Please.

But all of the methods ought to be equivalent, no? The only question is how much of the work is contained implicitly in the notation and how much is done explicitly.

 

[PeroK]

Please.

But all of the methods ought to be equivalent, no? The only question is how much of the work is contained implicitly in the notation and how much is done explicitly.

The Klingons would probably like this bit:

And enterprises of great pith and moment,
With this regard their currents turn awry,
And lose the name of action.

 

[itfitmewelltoo]

How can
I derive mathematically law of composition of
velocities from time dilation and length
contraction
But please use only algebra .Don't use four vectors
and space time diagrams.

" law of composition of velocities" I am not sure what that is.

 

[Doc Al]

" law of composition of velocities" I am not sure what that is.

It is usually called the "addition of velocities" formula.

 

[stevendaryl]

It might be obscure how it works, but an especially slick form of the velocity addition law is in terms of "rapidity":

Define an object's rapidity to be: $\theta = tanh^{-1}(\frac{v}{c})$, where $tanh^{-1}$ means the inverse of the hyperbolic tangent function. As $v$ ranges between 0 and $c$, the rapidity just keeps increasing from $0$ to $\infty$. In terms of rapidity:

If object $A$ has rapidity $\theta_{AB}$ relative to object $B$'s inertial frame, and object $B$ has rapidity $\theta_{BC}$ relative to object $C$s inertial frame, then $A$ has rapidity $\theta_{AC} = \theta_{AB} + \theta_{BC}$ relative to object $C$. Rapidities add in SR just like velocities do in Newtonian physics.

 

[Sorcerer]

All you have to do is divide the distance Lorentz transformation equqtion by the time Lorentz transformation equation.

x/t = u = γ(x’ + vt’)/(γ[t’ + vx’/c²])

Then divide by 1 in the form of t’/t’ and vaala:

u = (x’/t’ + vt’/t’)/[(t’/t’ + vx’)/(t’c²)]
u = (u’ + v)/(1 + u’v/c²)

 

[Herman Trivilino]

Thanks for answering the question. What do you mean ?is not time dilation and length contraction enough? Why?

Length contraction and time dilation follow from the two postulates, but so does relativity of simultaneity. You can derive the velocity composition formula from the postulates, what you cannot necessarily do is derive a couple of consequences from the postulates, and then expect to use those consequences in place of the postulates.