Squared numbers and square root (Need help with explaination)

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Discussion Overview

This discussion revolves around the mathematical concepts of squaring numbers and square roots, particularly in the context of physics equations such as "E=mc^2" and time dilation. Participants explore the implications of these mathematical operations in special relativity and their significance in physical equations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asks why the speed of light is squared in "E=mc^2" and seeks clarification on the meaning of square roots in equations like time dilation.
  • Another participant explains that squares and square roots arise from the derivations of special relativity, emphasizing their mathematical utility without asserting their inherent significance.
  • A participant requests further clarification on the relationship between square roots and absolute value, specifically referencing the substitution abs(x) = sqrt(x^2).
  • Discussion includes the idea that energy units require a quantity with dimensions of length²/time², which the speed of light squared provides, and that this fits into the framework of special relativity.
  • One participant notes that the mathematical framework of special relativity was developed for its effectiveness in describing physical phenomena, suggesting that verbal explanations may fall short.
  • Another participant mentions the relevance of root mean square (RMS) and the concept of areas in mathematical calculations as further examples of second-order relationships in physics.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and seek clarification on mathematical concepts, indicating that there is no consensus on the interpretation of these mathematical operations in the context of physics.

Contextual Notes

Some participants highlight the complexity of mathematical interpretations in physics, suggesting that a purely verbal explanation may not suffice. There is an acknowledgment of the patterns that emerge in mathematical expressions, but the discussion does not resolve the underlying uncertainties or assumptions related to these concepts.

Mulz
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Can anyone tell me why for example the speed of light is squared in "E=mc^2" ?
Also what does square root mean and why is it in certain equations like for example time dilation?

What happens if you exclude the square root and the y^x in a equation?

I am still studying high school physics, but it would be nice for someone to explain it.
 
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The square and square roots come into the equations as a result of their derivations from the basic premise of special relativity that the speed of light is the same for all observers.

Start by reading the Wikipedia article on special relativity to see how they come into play:

http://en.m.wikipedia.org/wiki/Special_relativity

In math square and square root are complementary operations like adding and subtracting or multiplying and dividing. There is nothing inherently special about them.

However they are very useful. As an example, square roots are used in calculus as a stand-in for absolute value.
 
jedishrfu said:
T There is nothing inherently special about them.

However they are very useful. As an example, square roots are used in calculus as a stand-in for absolute value.

Could you explain further? I confess I have no idea what that means.
 
sophiecentaur said:
Could you explain further? I confess I have no idea what that means.

My apologies I wasn't clear enough, I was thinking of the abs(x) = sqrt(x^2) substitution.
 
Units of energy are mass * length^2 / time^2. So, you need something with units of length^2 / time^2 to get a quantity of energy to match up with a quantity of mass. Speed of light squared has units of length^2 / time^2. It just happens that it fits into the equation.

You might ask why the speed of light is used here and not some other speed. The simple answer is that it matches experiment. But, the principles of special relativity can be derived from thought experiments in which we assume that the speed of light is the same for all observers. You can probably find an explanation based of light signals somewhere if you look hard enough. The Lorentz factor (with the square root) happens to be exactly what is needed to allow the light signals to work properly in any frame of reference. Hendrik Lorentz used this mathematics to explain the results of the Michelson-Morley experiment..
 
Mulz said:
Can anyone tell me why for example the speed of light is squared in "E=mc^2" ?
Also what does square root mean and why is it in certain equations like for example time dilation?

What happens if you exclude the square root and the y^x in a equation?

I am still studying high school physics, but it would be nice for someone to explain it.

This may sound like a bit of a cop-out but I have to say that the Maths of SR (and throughout Physics, am) was developed because it is a far better and more concise way of describing and predicting what goes on. People frequently ask for a 'Physical interpretation' of what the Maths means - perhaps when they are just starting on their Serious Maths education or possibly because they never took it further. In most cases, a strictly verbal argument is pretty much doomed to failure - which is why the Maths was brought into Science in the first place. If Newton hadn't invented his own personal version of the Calculus he would never have got where he did. The wording of even simple laws about the relationships of things just gets people bogged down when trying to explain things. Throughout Science, we come across the same, stock pieces of Maths - the same patterns.

It is good that you have noticed that these squares and square roots keep cropping up all over; you have spotted pattern. Once you have done some more Maths and, particularly, Maths in Physics, you will find that the Maths carries a useful message and that these patterns are a result of juggling round with the symbols and, particularly when Calculus is introduced.
 
jedishrfu said:
My apologies I wasn't clear enough, I was thinking of the abs(x) = sqrt(x^2) substitution.
Oh yes. I see where you're going.
RMS is another example of the second order coming into things. And those calculations involving of 'areas under a triangle', too.
 

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