Squared numbers and square root (Need help with explaination)

In summary, the speed of light is squared in E=mc^2 because it is the speed at which energy travels in a vacuum. Square roots come into equations like time dilation because they allow for quantities of energy to be matched with quantities of mass. Excluding the square root and y^x in equations can lead to errors in calculations.
  • #1
Mulz
124
5
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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  • #2
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
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.
 
  • #4
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.
 
  • #5
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
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.
 
  • #7
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.
 

Related to Squared numbers and square root (Need help with explaination)

1. What are squared numbers?

Squared numbers are numbers that are multiplied by themselves. For example, 3 squared is 3 x 3 = 9. The result of a squared number is always a perfect square, meaning it has a whole number as its square root.

2. What is a square root?

A square root is the inverse operation of squaring a number. It is the number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 x 3 = 9.

3. How do you find the square of a number?

To find the square of a number, you multiply the number by itself. For example, to find the square of 5, you would do 5 x 5 = 25.

4. How do you find the square root of a number?

To find the square root of a number, you can use a calculator or use a method called "prime factorization." This is when you break down the number into its prime factors and then take the square root of each factor. The product of all the square roots will give you the square root of the original number.

5. Why are squared numbers and square roots important?

Squared numbers and square roots are important in many fields, including math, physics, and engineering. They allow us to solve equations, find unknown values, and understand the relationships between numbers and geometric shapes. They are also used in real-life applications such as calculating areas and volumes, determining distances, and analyzing data.

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