How to Derive the Lorentz Factor from Pythagoras?

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The discussion revolves around deriving the Lorentz factor using Pythagorean principles, specifically through the equation involving a light clock on a train. A participant expresses confusion about isolating variables and manipulating the equation correctly. They struggle with moving terms around, particularly with the presence of multiple instances of t^2. Other contributors provide guidance on rearranging the equation and emphasize that the process is similar to solving simpler algebraic equations. The conversation highlights the importance of understanding algebraic manipulation to derive the Lorentz factor effectively.
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Hey guys, this is a little silly question but it bothers me. I am not a math genius (yet i hope) and I am still in elementary school so there's a lot to learn. But i just read about the lorenz factor in this example he basically used pythagoras of this light clock in a train, so it started of as

(ct)^2 = (cx)^2 + (vt)^2

and he derived it into:

t = \frac{x}{\sqrt{1-\frac{v^2}{c^2}}}

I would have posted an attemp to solve it but i really just don't know how to crack it and get started

Pleeeaaase help it would be really nice :D
 
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That is nothing you can derive (at least not in the way you ask for here), that is a definition of γ.
 
right its me god I am stupid! he solved for t not gamma don't really know what went through my head while i wrote it. i corrected it in the post now
 
OK, I'll help you get started: bring all terms which have a ##t## to one side of the equation and the other terms on the other side. Isolate ##t^2## so you have ##t^2 = \text{something}##. Then take roots.
 
ok ill try:

c^2t^2 = c^2x^2 + v^2t^2
t^2 = \frac{c^2x^2 + v^2t^2}{c^2}
Dividing both sides by t^2
1 = \frac{c^2x^2 + v^2t^2}{c^2t^2}

im stuck... lol
normally i don't really have trouble when solving for variables but this one irritates me.. can i have another hint ? :)
 
MathiasArendru said:
ok ill try:

c^2t^2 = c^2x^2 + v^2t^2
t^2 = \frac{c^2x^2 + v^2t^2}{c^2}

I can see a ##t^2## on the LHS and on the RHS. The idea is to have all occurences of ##t^2## on the LHS.
 
Exacly that's the though part, because normally i would just divide out the t^2 but that won't help in this example as it would leave me with a 1 on the LHS.. and that wouldn't help much,, is there some mechanism or method that i am missing that could solve this? i feel like there's something i haven't learned that could allow this to be solved.. or is it just me that's blind?
 
MathiasArendru said:
Exacly that's the though part, because normally i would just divide out the t^2 but that won't help in this example as it would leave me with a 1 on the LHS.. and that wouldn't help much,, is there some mechanism or method that i am missing that could solve this? i feel like there's something i haven't learned that could allow this to be solved.. or is it just me that's blind?

If you have ##2x^2 = 5 - 3x^2##, can you solve that? You just need to do the same thing here. Isolate ##x##.
 
Yea no problem but there i can just add 3x^2 to both sides, in my example its v^2t^2 so i can't do the same thing here?
 
  • #10
MathiasArendru said:
Yea no problem but there i can just add 3x^2 to both sides, in my example its v^2t^2 so i can't do the same thing here?

You could try subtracting ##v^2t^2## from both sides.
 
  • #11
MathiasArendru said:
Yea no problem but there i can just add 3x^2 to both sides, in my example its v^2t^2 so i can't do the same thing here?

Why do you think v2 is different from 3?
 

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