Proof regarding composition of velocities

[drnickriviera]

Homework Statement

This is a problem from D'Inverno's "Introducing Einstein's Relativity".

If v_AB is the velocity of B with respect to A, v_BC is the velocity of C with respect to B, and v_AC is the velocity of C with respect to A (all velocities are in relativistic units, that is, c=1), prove that if 0<v_AB<1 and 0<v_BC<1, then v_AC<1.

Homework Equations

The problem should be resolvable with just the equation

v_AC=(v_AB+v_BC)/(1+v_ABv_BC).

The Attempt at a Solution

I understand that there are other ways to prove this, but I want to know this particular approach. I suspect it boils down to showing that the numerator is less than the denominator. I have tried reducing the denominator and/or increasing the numerator to find a greater expression that is less than one, but so far nothing has worked. I know this is really just a mathematical proof and has little to do with conceptual relativity, but I'd still like the solution.

 

[Dick]

You want to show (a+b)/(1+ab)<1 if 0<a<1 and 0<b<1, right? Write it as a*(1-b)+1*b<1. a*(1-b)+1*b is a linear function of b as b goes from 0 to 1, right also? It's a weighted average of a and 1. So it must hit it's max and min at b=0 or b=1.

 

[drnickriviera]

You want to show (a+b)/(1+ab)<1 if 0<a<1 and 0<b<1, right? Write it as a*(1-b)+1*b<1. a*(1-b)+1*b is a linear function of b as b goes from 0 to 1, right also? It's a weighted average of a and 1. So it must hit it's max and min at b=0 or b=1.

I'm sorry, I don't really understand how those two (the original expression and the linear function) are equivalent. Could you explain a little more deeply?

 

[hikaru1221]

Since a<1, b<1, we can rewrite them as: a=1-x, b=1-y, where x>0, y>0. This will make it a lot easier

 

[drnickriviera]

Ok, that works. Thanks a lot!

 

[Dick]

I'm sorry, I don't really understand how those two (the original expression and the linear function) are equivalent. Could you explain a little more deeply?

(a+b)/(1+ab)<1. Multiply both sides by (1+ab). (a+b)<1+ab. Subtract ab from both sides. a+b-ab<1. Collect terms and factor. (a-ab)+b<1, a*(1-b)+1*b<1.