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"How do scientists derive equations?" is a huge question, whos answer spans thousands of years of history, and will differ based on who you ask. At its most basic level physicists use equations to come up with a general framework to describe the phenomena they observe. For instance consider $F=ma$, what we are trying to describe here is the ...

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We derive equations of formulas depending on our needs to understand a system. Some systems are good to be understood with just working with force, some may require other quantities to properly describe what is going on in the system. Why only multiplication and not addition? Simply because you can multiply two different units but adding them doesn.t make ...

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But if we measured light in metres, wouldn't we get joules since we are using SI units? Of course we would get joules. It's the SI unit of energy. But in atomic physics one often uses a custom unit, for convenience, and this unit is just what it's called: electron [charge times] volt (by charge we mean its absolute value). If you do the calculation, you'll ...

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It isn't always measured in nm. Why would you think that? We often pick units to make the number easier to handle, e.g. bring the typical measurements in to the range of 1's, 10's 100's which people can comprehend. Visible light is in the range of 100's of nm, and the size of an atom approximately 0.1 nm. And we ID elements, and bonds in compounds, from ...

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One often arrives at delicate points when trying to find a literal connection between mathematical and physical approaches to problems. As G. Smith already commented, the important aspect here is modeling. Roughly, this goes in three steps: You have a real (physical) system and you map it to some mathematical structure. You analyze the properties of the ...

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If $apple$ is a valid unit then $apple^2$ is also a valid unit. Simply because it is valid does not imply that it is meaningful. Whether or not a unit is meaningful has to do with whether it is used in any physical formulas. I don’t know of any formulas that involve the unit $apple$, nor any that take products of quantities measured in $apples$. So to my ...

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The thing missing from the units given in examples (m, C, etc) in multiple dimensions is a qualifier of which dimension. For example, when we identify an area and give it a unit, say $m^2$, we really mean $m$ in direction x times $m$ in direction perpendicular to x. When we say $C^2$ (Coulomb), we mean $C$ of one entity times $C$ of a different entity. These ...

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Simple, it's $3\times 4 = 12 ~\text{apple}^2 \equiv \text{Area}$ in terms of apples : Definition of area doesn't changes if you use $m^2$ or $cm^2$ or $\text{apple}^2$, or anything else. So to say, mathematical expression is invariant of the units used at hand. One can even say that $t^2$ is hyper-surface in time. No problems at all.

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The Lorentz transformation where we consider motion in the x-direction only, is given by the equations $$\tag 1 x’ = \gamma (x - vt)$$ $$\tag 2 t’ = \gamma (t -\frac{vx}{c^2} )$$ $$y’=y$$ $$z’=z$$ where $$\tag 3 \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$ The question is, can these equations $$x' = 1.25x - 0.75t$$ $$t'=1.25t -0.75x$$ $$y'= y$$ $$z' = z$$ ...

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The metre is defined as the distance light travels in a vacuum in 1/299792458 of a second. This is totally independent of temperature, or indeed the length of any physical object. From the same source, you can see that in the past when the metre was defined in terms of a physical object (a platinum-iridium bar), a precise temperature and pressure were ...

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