How do scientists derive equations?
For example, how do they know that:
Work done = Force × Displacement
And why is it always multiplication and not anything else? Why not addition?
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1$\begingroup$ Are you primarily interested in the derivation of work? If so, maybe you should remove the “for example” $\endgroup$– DaleJan 24, 2021 at 4:07
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$\begingroup$ I don't get it why you say- "And why is it always multiplication and not anything else, why not +?" Haven't you seen equations like- $$S=ut+\frac{1}{2}at^2$$ They indeed have '+'. $\endgroup$– leeJan 24, 2021 at 6:52
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2 Answers
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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 conservation of momentum, that is to say, left untouched a system retains its momentum ~ $a=0$, and a force is a "thing" which is done to a system to change its momentum $F=\frac{dp}{dt}=m \frac{dv}{dt}=ma$.
From this law we can try to see how the Kinetic Energy of an object will be affected by a force acting on it, and as it so happens this change in kinetic energy is proportional to the distance at which the Force is applied, I will link some derivations for this, but:
$$\Delta E = \int_d F \cdot dx$$
We then define that change in Energy of the object as "Work Done To Object", and when we are on a straight path with constant force in the direction of the path, $W=Fd$.
Learning where these equations come from is a pretty phenomenal thing, I recall always wondering why it is $E=mc^2$ why not $E=mc^{2.0001}$ or $E=mc^{1.999}$, where do these units come from? Understanding that Physics is not simply having equations that work, but instead developing equations in a way that describes reality greatly helps.
answered Jan 24, 2021 at 4:25
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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 sense. For example, lets take length and area. Multiplying them gives volume but adding them doesn't make any sense. Addition is sensible if the added quantities are the same, and they add to the same kind of quantity.