Equations like E=mc^2 and stuff like how the force og gravity becomes 1/4 as strong when the distance between the objects dobles. Similarly with braking distance with cars, double the speed and the braking distance quadruples. These all seem to fit so well.
Have we made math to fit so nicely with physics? Am I thinking of all this wrong? Since I feel it like it would be to big of a coincidense that we can so easily use equations to predic physics. What is actually the reason for this?
In: Mathematics
Many (most?) famous equations a*ren’t* perfectly convenient.
The foundation of these equations are just simple relationships between things we find useful to define. F=ma just means that double the mass requires double the force to accelerate, or the same mass requires double the force to accelerate twice as fast. each side increases proportionally to each other. The fact that m & a and being multiplied together on that side signifies that increasing either of them increases force by a proportional amount. F=ma is nice because each side is proportional 1:1.
**The bigger part of this** is that many (most?) famous equations a*ren’t* perfectly convenient. the ‘c’ in E=mc^(2) is the only hard number in there, and it’s about 299,792 km/s, and it is the number that tells you exactly what the ratio between the two sides of the = is. 1 ‘unit’ Energy is equivalent to 89,875,243,264 ‘units’ mass. Hardly a convenient ratio.
Similarly, the equation we use to calculate the exact force of gravity between two masses is F = G([mass1*mass2]/distance^(2)). The d^(2) is tells us gravity falls off at the square of the increase in distance (2x distance = 1/4 force, 3x distance = 1/9 force, which is because distance is one dimension but gravity spreads out over 3 dimensions; this is true of ectromagnetic fields too). G, however, is the gravitational constant, the only hard number in the equation, and is about 6.67430(15)×10^(−11) m^(3) * km^(-1) * s^(-2). This is, obviously, a weird inconvenient number, but it shows the exact ratio between each side of the equation for the units we created.
TL;DR most famous physics equations record proportional relationships between things, and it’s very common for one thing to double if another thing doubles (like, mass and force, or acceleration and force). “2x = 1/2y” and “2x = y^(2)” are also common in nature. There are relationships in nature that are mathematically more complicated than this, but the most common and most fundamental relationships are often the simplest. That just seems to be the way the universe works.
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