Yeah I know it’s an odd question, but I have wondered why so many physics formulas for describing physical interactions and relationships use squares all the time. For example E=MC[sup]2[/sup]. Is it simply a function of how equations need to be written or what? Why do so many physical descriptions of the real world require a number to be multiplied by itself?
I think it’s because so many physical interactions we care to model have a time component (the first that comes to mind is 1/2 mv[sup]2[/sup]), making the calculus useful. For example, when we take the derivative with respect to time (velocity, technically) we get momentum (mv). So, naturally, when we take the integral of momentum, we’re going to get a squared term.
Any case where we’re switching from one to two dimensions is going to have a squared term as well (pi r[sup]2[/sup]), which explains why it pops up in so many trig formulas sin[sup]2[/sup] + cos[sup]2[/sup] = 1 (think unit circle).
ETA: Parabolic trajectories are another time when the squared term naturally appears.
dx[sup]2[/sup]/dt[sup]2[/sup] = dx/dv = a
dx/dt = v = a*t + v0
x = 1/2 a * t[sup]2[/sup] + v0 * t + x0
Not quite as erudite as the previous post, and perhaps a bit of restatement, but much of physics is based upon field dynamics, and most of those are goverened/described by the inverse square law. In performing dimensional analysis of any equation to bring it to its simplest form, the inverse is always represented of the square of the dominant variable.
FML
And in layman’s terms the inverse square law is in itself has to do with relationship of distance to area. If you imagine a point source of uniform anything – if you put a piece of paper to block that anything it will block a certain % of it determined by the paper’s surface area. If you move it twice as far away from the source, you will be blocking four times less of that anything.
I’ve often wondered about this, but IANAM(athematician), so could this be explained in words of less than one syllable, and using no number greater than 1? 
(or at least, as simply as possible). Many thanks.
Imagine a light bulb. Imagine a black piece of paper one foot in front of it. It’s blocking a certain amount of light. If this piece of paper was a sphere with the light bulb in the center, it would block ALL the light. Now, since it’s not, it’s blocking part of the light which is roughly the ratio of the area of the piece of paper to the area of this total sphere of one foot radius.
If you move it twice as far from the bulb, it will block four times less light. This is a geometric fact because the area of the piece of paper didn’t change, but the area of this virtual now 2-foot radius sphere is four times bigger than in the 1-foot radius case.
In other words, there is a lot of squares in physics because a lot of relationships are directly or indirectly related to the relationship between distance and area. For example, if you graph the curve of acceleration of some object, and then calculate the area underneath the curve you are going to get the objects speed.
Og have thing. Thing is this far from light. Thing is bright. Now Og move thing this far one more time. Thing less bright. How much less bright? Og use math, find out! It now one in one plus one plus one plus one less bright. Since Og move one plus one, must be square law flip over!
Brilliant!
Can I add another related question here, rather than start my own thread? What’s the highest power to appear in an equation describing a scientific law? Are there any laws involving, say, a fourth power? A fifth? Even higher powers?
The Stefan-Boltzmann Blackbody Radiation Law has a fouth Power:
The Richardson-Dishman equation for Thermionic Emission has both a second power and an exponential in the same variable, temperature, so it’s not a higher power, but it is complicated:
I also note that several empirical relations (not"physical laws") frequently use higher powers – sometimes much higher powers. The lifetime of a lamp filament, for instance, goes as the twelfth power of the current or voltage.
Presumably that’s the *inverse *twelfth power. This is the sort of thing I had in mind, though. Thanks!
Now’s as good a time as any to point out to everybody that x[sup]y[/sup] is easy to code. Use sup inside square brackets for the open tag and /sup inside square brackets for the close tag.
Oh, and how could I forget a classic example from optics – the cos[sup]4[/sup] law of Irradiance.
The Hazen-Williams formula for determining pressure loss in a section of pipe has a ^4.8655 in it:
f = 0.2083 (100/c) * 1.852 * q^1.852 / dh^4.8655
f = friction head loss in feet of water per 100 feet of pipe
c = Hazen-Williams roughness constant
q = volume flow (gal/min)
dh = inside hydraulic diameter (inches)
Not quite a physical law, but I think it meets the intent…
Hey now! I can swing a sledgehammer with the best of em. 
It occurs to me, as well, that frequently you only have powers up to the square because in many cases the theory assumes linear relations. In the area of Nonlinear Optics, for instance, one treats of interactions between the electromagnetic fields of light beams and the dielectric properties of materials that occur when the interactions are not linear, and you can get higher-order terms appearing. Similar cases occur elsewhere as well. For my thesis I treated the next order of correction terms for rotational energy, involving fourth order terms.
Electromagnetic interactions (and gravitational ones, too, for that matter) can be treated in terms of quadrople and higher order terms that go beyond inverse square terms. I once worked on v-v transitions of trapped species that varied as the inverse sixth power of separation.
Friction and drag are frequently treated as linear, but that’s a simple approximation, too. You really need to include higher orders for larger ranges. (In c.S. Forester’s “Admiral Hornblower in the West Indies” one young officer eagerly points out how drag varies as “the secord order or even higher power of the speed”). But maybe that falls more into the range of “Empirical relations” than Physical Laws.
Drag is a tricky one,** Cal**, you’re right. One way to get around using higher orders for larger ranges is to cut your range down and vary your drag coefficient instead. It’s been awhile since I’ve looked at a drag coefficient vs. Mach number plot, but IIRC it goes up fairly linearly to about .9, then drops off until about 1.1, and then levels off at about 80% of the max value.
To take this into account in the past for a situation where the velocity of the projectile was going to be changing (it was being shot as vertical as possible), I wrote my MatLab code with a lookup table that used the proper expected drag coefficient based on the current expected velocity. So, although the equation was only second order, it probably could have been put in the form of a much higher order if a constant drag coefficient.
Rutherford scattering also has a trig function to the fourth power in it.
E=MC[sup]2[/sup] is only the special case of the relativistic energy-momentum relationship.
E[sup]2[/sup] - (pc)[sup]2[/sup] = (mc[sup]2[/sup])[sup]2[/sup]
That’s a fourth degree right there.
To go back a step, though, remember that what the equation really says is that mass and energy are the same thing but manifest differently. The real equation is :
E ~ M (The proportional symbol from the character box doesn’t work for me, so pretend that the tilde means proportional to.
A little bit of energy manifests as a little bit of mass. A lot of energy manifests as a lot of mass. To turn the proportionality into an equation a constant must be added.
E = MConstant
How do you express the constant? You do so in a way that makes the units correspond.
Energy is work, defined in the SI system as joules, which are equivalent to newton-meters. A newton-meter is kg*m[sup]2[/sup]/s[sup]2[/sup]
So any constant must have the units so that C = E/M
kg*m[sup]2[/sup]/s[sup]2[/sup] / kg.
C = m[sup]2[/sup]/s[sup]2[/sup].
This is obviously a speed squared. It can’t be anything else. Einstein’s insight was the the speed involved was the theoretical speed of light in perfect vacuum. That’s why C is sometimes called Einstein’s Constant. The squaring results from the original definition of work. Work is a force that can cause an object to accelerate. You can derive E = MC[sup]2[/sup] from F = MA. Acceleration is a second derivative. And that brings us right back to post #2, completing the circle.
What words are those?