Re: Stuck - Probably simple algebra

From: Dirk Van de moortel (dirkvandemoortel_at_ThankS-NO-SperM.hotmail.com)
Date: 12/24/04 

Date: Fri, 24 Dec 2004 12:48:54 GMT

"Tom Capizzi" <tom.capizzi@verizon.net> wrote in message news:8JNyd.11229$EL5.824@trndny09...
>
> "Dirk Van de moortel" <dirkvandemoortel@ThankS-NO-SperM.hotmail.com> wrote
> in message news:VSIyd.8698$Gd4.71751@phobos.telenet-ops.be...
> >
> > "Zinc Potterman" <zincnews@tiscali.c123o.uk. (delete digits to reply)>
> > wrote in message news:cqffnn$lpk$1@hercules.btinternet.com...
> >> I'm using D'inverno's, Introducing Einstein's Relativity to try and self
> >> teach.
> >> I'm stuck with what will probably turnout to be obvious to someone.
> >>
> >> The text uses the k calculus to derive the composition law for velocities
> >> in
> >> Sp Rel
> >>
> >> Eqn(2.5) ksubac = ksubab ksubbc
> >>
> >> is used with
> >>
> >> Eqn (2.4) k=root(1+v) / root(1-v)
> >>
> >> to derive
> >>
> >> Eqn (2.6) vsubac = (vsubab + vsubbc)/ (1 + vsubab vsubbc)
> >>
> >> I tried direct substitution of each corresponding k value and it just
> >> became
> >> very messy
> >> and looked to be going nowhere towards 2.6
> >> Thanks
> >> K
> >
> > I don't have the book, but here's how to go about it, using some
> > shorter notation:
> > K = ksubac
> > L = ksubab
> > M = ksubbc
> > w = vsubac
> > u = vsubab
> > v = vsubbc
> >
> > You have
> > K = L M
> > therefore
> > sqrt(1+w)/sqrt(1-w) = sqrt(1+u)/sqrt(1-v) sqrt(1+u)/sqrt(1-v)
>
> typos here! should be: = sqrt(1+u)/sqrt(1-u) sqrt(1+v)/sqrt(1-v)

Bad copy paste job with the LHS :-)

Dirk Vdm

>
> > square to
> > (1+w)/(1-w) = (1+u)/(1-u) (1+v)/(1-v)
> >
>
> At this point it is easier to simply collect terms, rather than
> introduce another substitution:
>
> Multiply both sides of the equation by (1-w)(1-u)(1-v):
> (1+w)(1-u)(1-v) = (1+u)(1+v)(1-w)
> 1 + w -u -v -uw -vw +uv +uvw = 1 -w +u +v -uw -vw +uv -uvw
> 2w + 2uvw = 2u +2v
> w (1 + uv) = u + v
> w = (u + v)/(1 +uv)