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Asked by TylerC on 3 Jun 2024.
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Tim Chapman answered on 3 Jun 2024:
Hi Tyler, if I remember back to my A Level Maths, and it’s a while ago, this is because that’s basically how geometry works.
A square is a common and easy shape. A right-triangle is half a square. A circle is related to a square via π, or to a right-triangle via sin/cos. Coordinate systems are either based on squares or circles.
Almost every 2D geometry formula will involve a square. And a huge number of physics problems will have squares in for the same reason.
Hope this helps 🙂
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Sheridan Williams answered on 3 Jul 2024:
What a great question. I asked AI and it said:
Symmetry and Simplicity:
Squaring the speed of light (c^2) maintains symmetry.
It aligns with the quadratic kinetic energy term.
Simplicity and elegance are prized in scientific equations.I’m afraid that it doesn’t answer the question why it’s not to the power of 1.5, 2.5, or even cubed.
Squaring something is sensible with areas etc.I’m going to have to think about this, but will watch for other answers from scientists.
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Tom Ranner answered on 3 Jul 2024:
For each formula there is a different answer. There is a precise answer for why the square of something comes in. Usually these derivations require lots of experiments and hard thinking to get to the right answer – fortunately for you someone has done those for the equations you have written down already!
There is another way to think which is a useful check for any scientific formula – that’s to use units. Units describe how a number relates to the real world quality. For example, 1 metre is very different from 1 millimetre which is very different for 1 kilometre.
When you write down a formula like E = m c^2 then you are saying that the units of the thing on the left have to match the units of the thing on the right. Energy (E) is measured using Joules, mass (m) is measured using kilograms and the speed of light squared (c) has units metres per second. So the formula says that a Joule should be the same as one kg times metre^2 per second^2, which is correct.
This means that we know the power of c in this formula can only be 2 and no other value is possible!
Comments
Luke commented on :
In the case of kinetic energy (E_k = 0.5 m v^2) note that momentum p = mv, so you could also write E = mv^2 = pv = momentum times velocity. So the squared symbol here is due to how the velocity comes up twice in the product. Once directly and again via the momentum.
Actually, from memory, I think the energy E=mc^2 does actually derive from the kinetic energy of a photon, though it’s been a while since I had a lecture on that subject! If you study physics at university, you will probably do a course of special relativity where it will come up.
I’m not sure if this is what you were asking, but the reason something timesed by itself is called a “square” is basically because the area of a square is the length of an edge timesed by itself (area = length x length = length^2) but it doesn’t always mean a physical square shape is involved in the situation being described.