# Squared numbers and square root (Need help with explaination)

1. Nov 4, 2014

### Mulz

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.

2. Nov 4, 2014

### Staff: Mentor

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.

3. Nov 4, 2014

### sophiecentaur

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

4. Nov 4, 2014

### Staff: Mentor

My apologies I wasn't clear enough, I was thinking of the abs(x) = sqrt(x^2) substitution.

5. Nov 4, 2014

### Khashishi

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..

6. Nov 4, 2014

### sophiecentaur

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.

7. Nov 4, 2014

### sophiecentaur

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.