When is it ok to use non-relativistic equations?

[jaketodd]

I saw here on Wikipedia that if the speed you're working with is less than 10% the speed of light, Newtonian Mechanics are still accurate. So, my question is: If you are working with a speed that is less than 10% the speed of light, then will all Newtonian Mechanics variables and equations still hold true? Also, what does it mean on that page by "within a margin of error of 1%"? Does it mean that if you are working with a speed that is 10% the speed of light, then your Newtonian Mechanics calculation will be accurate to within 1% of your figures? What if you are working with a speed that is 5% the speed of light - is the margin of error still 1%? Or, by margin of error, does it mean that Newtonian Mechanics hold if your speed equal to, or below 10% the speed of light +/- 1% of the speed of light?

Thanks,

Jake

 

[jedishrfu]

I think it means that if you do a calculation say for the momentum of the speeding object in Newtonian terms vs Einsteinian terms they will differ by one percent. If the speed is cut in half the they will differ by half a percent.

 

[jaketodd]

Thanks jedishrfu.

Anyone else want to confirm this?

Thanks,

Jake

 

[Michael C]

A simple example can help. If we are comparing two velocities in the same direction, it's easy to calculate the difference between the values given by Newtonian mechanics and by Special Relativity (velocities shown as fractions of c, the speed of light):

In Newtonian mechanics, the relative velocity of particles moving with velocities v and u (in the same direction) is v - u.

In Special Relativity, the relative velocity is (v - u)/(1 - vu)

The two results differ by a factor of 1 - vu. If v and u are both 10% of c, then the difference between the Newtonian result and the one predicted by SR is 1%, the value given by the Wikipedia article. If v and u are both 5% of c, then the difference between the two results is only 0.25%.

You can see that the error margin doesn't increase linearly with the speeds, but exponentially.

 

[Lsos]

Note that depending on desired precision, sometimes even at much lower speeds we need to use relativistic equations. The often-cited GPS satellites is one example, where even an error of a tiny fraction of a percent is unacceptable.

 

[Dale]

Does it mean that if you are working with a speed that is 10% the speed of light, then your Newtonian Mechanics calculation will be accurate to within 1% of your figures?

Not necessarily. Your distances and times will be accurate to within 1%, but depending on the details of your calculations that 1% error can cause much larger errors. For example, if you are subtracting two large distances which are very close, then the 1% error in each distance can lead to a much larger error in the difference. This is essentially why relativistic corrections need to be made for GPS as mentioned by Lsos.