When is it ok to use non-relativistic equations?

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In summary: So, in summary, if your speed is less than 10% the speed of light, Newtonian Mechanics can still provide accurate results within a margin of error of 1%. However, this margin of error does not increase linearly with the speed, and depending on the details of the calculations, it can cause larger errors. This is why relativistic corrections are necessary, even at lower speeds such as those used for GPS satellites.
  • #1
jaketodd
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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
 
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  • #2
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.
 
  • #3
Thanks jedishrfu.

Anyone else want to confirm this?

Thanks,

Jake
 
  • #4
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.
 
  • #5
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.
 
  • #6
jaketodd said:
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.
 

1. When can I use non-relativistic equations in my experiments?

Non-relativistic equations can be used when the objects involved are moving at speeds much slower than the speed of light. This is typically the case in everyday experiments and observations.

2. Are there any specific conditions where non-relativistic equations are not applicable?

Non-relativistic equations are not applicable in situations where the speeds of objects are close to the speed of light, such as in high-energy particle collisions or near massive objects like black holes.

3. How do non-relativistic equations differ from relativistic equations?

Non-relativistic equations are based on classical mechanics, which do not take into account the effects of special relativity. Relativistic equations, on the other hand, take into account the effects of time dilation and length contraction at high speeds.

4. Can non-relativistic and relativistic equations be used together?

Yes, in some cases, non-relativistic and relativistic equations can be used together to accurately describe a situation. For example, in the field of quantum mechanics, non-relativistic equations are used for slow-moving particles, while relativistic equations are used for high-energy particles.

5. Is it necessary to use non-relativistic equations in all scientific experiments?

No, it is not necessary to use non-relativistic equations in all scientific experiments. They are most commonly used in situations where the speeds involved are much lower than the speed of light. In other cases, such as in astrophysics or high-energy physics, relativistic equations are necessary for accurate results.

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