relative speed formula

[Didymus]

Doing a project in excel to make a little form to plug in linear speeds and have it spit out some data on the speeds estimated by SR and all that... And working with cells instead of normal variables made me entirely forget the formula... What part did I miss here:

(V+u)/(1+(vu)/c^2)

... What part did I screw up?

Edited October 12, 2013 by Didymus

[ajb]

This maybe better of in physics, but anyway your velocity addition formula looks correct, appart from the fact you have an odd number of braces. Braces always come in an even number...

[imatfaal]

Doing a project in excel to make a little form to plug in linear speeds and have it spit out some data on the speeds estimated by SR and all that... And working with cells instead of normal variables made me entirely forget the formula... What part did I miss here:

 

(V+u)/(1+((vu)/c^2)

 

... What part did I screw up?

 

excel isn't really a maths or physics tool - it's more business, accounting and layout; but it can do some great stuff. One thing really worth learning if you plan to make any formulae in excel is to name cells.

 

in your top left make a table of values column one name (no spaces no special characters) column two your value column three units

 

then high light first two columns and click create names - then choose "from left most column)

 

you have now named each of the value cells with the names. you can then write formulae like this

 

=(Vel_a-Vel_b)/(1-((Vel_a*Vel_b)/SoL^2))

 

when formulae get complicated it is invaluable being able to read them rather than have to refer back to anonymous cell references.

 

back to your formula

 

I would prefer to see it with minuses per mine above. Otherwise you get the relative velocity of two objects travelling at same speed in same direction as double - whereas they have zero relative velocity. Remember velocity is a vector

[Didymus]

1: You're right. Fixed the bracket problem. Thanks.

It's not for anything "useful" ... I've just never liked the theory, so I'm looking for a different way to visualize different circumstances without having to do math over and over. Here's what I put in:

(speeds in the same unit of measurement, obviously)

Speed 1 is in cell c2

Speed 2 is in cell c3

speed of C is in cell c4.

C5 shows the linear speed for reference: =C2+C3

C6 should show the speed according to einstein's dookie theory: =(C2+C3)/(1+(C3*C2)/(C4*C4))

C7 shows the correlation between the two: =C6/C5

(then a separate column to do the same thing converted into different units of measure)
kinda a cheat sheet to play with to help me visualize how much different parts really change when each other part changes.

 

Now I'm trying to figure out how to stick the lorentz transform in there in a meaningful way. Something like: =1/(sqrt((C4*C4)-(C6*C6))/C4)

... should show the factor by which time is thought to compress based off the total adjusted speed... then I could compare that to the same thing for the individual speeds in c2 and c3 to try to look at how one object can experience time at different rates relative to different objects

Edited October 12, 2013 by Didymus

[imatfaal]

use velocities (and minus in formula) not speeds - otherwise 5mph north and 5 mph north have a relative of 10mph and that is clearly wrong.

 

if you wanna visualize why not put everything in same cell? v1-v2/(1-v1v2/c^2) divided by v1-v2 your idea of relative speed and get the comparison in each cell of a table (remember A$4 copied will only change the column reference, $A4 will only change the row and $A$4 will copy with now changes)

[Didymus]

For simplicity, I'm assuming Instantaneous speed along a straight line. Less variables than trying to account for anything for which you'd need to specify "velocity." No changing directions or calculating vector or acceleration for now.

I'll definitely plug your thing in there... to clarify your variables though, are v1 and v2 referring to the individual speed of the two objects or the difference between linear speed and dilated speed? I'd assume the two objects except you're subtracting them... but if it were the other way around I don't follow why you'd multiply one by the other. Part of it looks like the lorentz transform... except I thought you had to take the square root of (C^2-v^2)... so I'm looking at something wrong... which...

oh my. I should go to bed.

[imatfaal]

go to bed!

 

v1 and v2 are velocities. in the newtonian model the relative velocity is v2-v1 . This accounts for direction as they are vectors - makes life much easier than trying to remember what direction something is heading.

 

the newtonian model

[latex]v_{21} = v_2-v_1[/latex]

 

fails at high speeds close to c - the new relavistic formula is

 

[latex]v_{21} = \frac{v_2-v_1}{1-\frac{v_2v_1}{c^2}}[/latex]