What is the threshold velocity for using the relativistic energy equation?

[Drizzy]

Homework Statement

To calculate kinetic energy I can either use E=mv^2/2 or for higher velocities I can use E=gamma factormc^2 - mc^2.

So my question is at which velocity can I use E=gamma factormc^2 - mc^2? Is 1000000m/s a high enough velocity?

Homework Equations

The Attempt at a Solution

 

[BvU]

Once $\gamma$ starts to deviate from 1 you will need the relativistic expression. Simple, isn't it ?

 

[Drizzy]

yes thanks a lot!

 

[BvU]

Just to get an idea: how much does $\gamma$ differ from 1 at the speed you mentioned ?

 

[Drizzy]

I checked it and it doesn't differ :P

 

[BvU]

But it does differ! perhaps not on a cheap calculator, but $\gamma = 1.0000055556... \ne 1$
What you mean is that it doesn't differ significantly...

 

[Drizzy]

yes well my graph calculator show that the gamma factor is equal to one :) But I get the idea

 

[BvU]

Sometimes helping the calculator a little by hand is more accurate: $$
\gamma = {1\over \sqrt {1-\beta^2}} \approx 1+{1\over 2} \beta^2 \, , $$ in your case $1+ {1\over 2}{1\over 300^2} = 1+ 1/180000 \ne 1$

 

[Ray Vickson]

Once $\gamma$ starts to deviate from 1 you will need the relativistic expression. Simple, isn't it ?

Actually, the issue is whether $\gamma$ differs significantly from $1 + \frac{1}{c^2} \frac{1}{2} v^2$, or whether $c^2 (\gamma - 1)$ differs significantly from $v^2/2$. For lab-scale speeds, $\gamma$ will hardly ever differ "significantly" from 1.