I am sorry, give me a minute.
…

You see, I easily lose my head. So, I need to use medications to balance my mood. Oh my God! That was supposed to be a secret, and it just slipped from my mouth to an admirer. Look, I suspect that once you learn my most-hidden aspects, my reputation with you will be ruined, and you won’t love me anymore. And it really distresses me to have failed your high expectations of an equation like me.

Nevertheless, it is such a heavy burden to suppress one side of mine while constantly showing the other. Yes, being an equation, I am well balanced and very clear about my views. I am one of those lucky theories that have a sound mathematical foundation, and this is why even the psychologists could not suspect a grain of problem due to my childhood. But I am telling you: I am bipolar….

Well, wait a minute! Who are you, and why would I share my secret with you? Go away with your own business, and leave me alone. I mean, I don’t want to disrespect you, but this is my private life, you know! …

No, no, wait, wait… Give me a second… Let me take my medication!

I know it sounds ridiculous if an equation complains of a disorder. How can you be an equation if you have a disorder, and how can you be in disorder if you are an equation? Well believe it or not, that’s exactly what I have been suffering from for such a long time. I am both orderly and disorderly; just like light is both wave and particle at the same time….

Hey, why are you looking at me like that? I told you to leave me alone. It was you who insisted on waiting and learning my secret. I don’t care if your designs are ruined or whatever. Actually, I’m the one who should be blamed. Why did I trust a person whom I met just a few minutes ago? I am like a fish that is eager to eat the worm on the hook. Such an idiot, I am…

Hello? Are you still there? I think I am losing my head again. Where are my pills?

Look, this is not easy for me to do, because in my routine life, people are obsessed with my equation aspect. And, they are passing this obsession from generation to generation. Every time I meet a young brain, my dream to finally meet an unconditioned mind fails. I am always too late to reach the fresh and eager minds. People always manage to find them before me, and instruct them to treat me as a well-established equation. No matter how much I want to show otherwise, they refuse to see my imperfect side, even if the experiments hit them in the face.

Why do they keep doing it? Much like a missionary, they are converting people into the belief of the Bernoulli Equation. I am telling you guys: I am bipolar, and I have imperfections, too!

What am I doing?! You are not going to believe me anyway. I am telling you my deepest secret, and making myself vulnerable as can be, but I still can’t wake you from a dream. You don’t want to wake up anyway; why would you? Do you have any other theory to believe in? Do you have another equation with which to orient yourself in an ocean of unknowns? Go on with your sleep, have nice dreams…

Unfortunately, I don’t have the same luxury as you. My continual tumbling between the two aspects of my reality never allows me to dream… Sometimes, I feel so energetic, so elated. I feel part of everything in the universe, and fall into an ecstasy by realizing that I contain everything in me. And the flow of time stops; we all become one and at peace with each other. No need to rush anywhere, no need to do anything… Just be…

But then my other side kicks in. I feel depressed under the burden of my duties and deadlines. I find myself in an ever faster pace of life, where there is no time to “just be”. There is always an action being commanded; there is always a motivation behind exchanges with others. The universe appears to be made of distinct individuals, like broken pieces of glass. Each is headed to a target of its own, unaware of any union. When life is filled with such a merciless momentum, what is more meaningful than getting rid of my life altogether?

Then, in that gloom, as my dizziness fades, my energetic side starts shining. By virtue of having been created in the same story, I focus on myself to read the universe. I realize that I carry the traits of anything and everything else around me. Once again, I start breathing the life that is gushing forth from me.

As I lose my conscience by getting high in life, I get stuck with the fact that losing my conscience defeats the purpose of being whole. You cannot hold onto the entities whose existence you fail to recognize. I start criticizing myself for disrespecting those around me. I blame myself for being such an addict and a loser; the shame of creation…

You see, there is no end to my ups and downs. As a remedy to these continuous head losses, I am using medications, but they have side effects. I am constantly growing fat, and losing my beauty. I hope one day, someone is going to be inspired with another equation that can take over my duties. Then I can retire from this bipolar life, and be mentioned in the scientific stories as a respectable grandparent…

The Bernoulli Equation given in its simplest form contains a pressure term (P) and a velocity term (1/2 (pV)^2), the sum of which is constant along a streamline:

P+1/2 pV^2=P_0

Thus, pressure and velocity work inversely with each other. As one increases, the other decreases.

The Bernoulli Equation can be derived in two independent ways. First, one can start with the conservation of energy (1st law of thermodynamics), and then follow some assumptions to end up with the Bernoulli Equation. Energy has a scale but no direction, and can be transformed from one form to another. Therefore, everything in the universe can essentially be converted into each other, just like the pressure and velocity terms in the Bernoulli Equation.

The second way to reach the Bernoulli Equation is based on the momentum equation (Newton’s second law). Again using few assumptions, one can achieve the nice and simple Bernoulli Equation. Unlike energy, momentum has both scale and direction. Any changes to this directionality require a forceful interaction between two objects.

Having its base on two fundamental laws of physics, hence reflecting both energy and momentum, makes the Bernoulli Equation one of the strongest and most beautiful equations of science, but not perfect!

The underlying theory of the Bernoulli Equation requires that the density (p) of the fluid be constant – in other words, incompressible. Therefore, the Bernoulli Equation fails to accurately explain the flow of gases at high speeds, which involves density changes. Similarly, the effects of viscosity are neglected in the foundations of the Bernoulli Equation. This means that the variation of pressure and velocity near a solid object cannot be explained by the Bernoulli Equation.

As a result of these imperfections, the Bernoulli Equation describes a fluid as a pile of metal sheets, and explains a flow as the sliding and bending of these metal sheets around each other. Clearly, this is not what a fluid is and not what happens in a real flow. Therefore, predictions of the Bernoulli Equation are inherently wrong, especially when there are sharp turns or obstructions in the way of the fluid.

In order to compensate for some of these deficiencies, a concept known as head loss (hloss) is introduced. Head loss is experimentally measured and then artificially added to the Bernoulli Equation, which then becomes:

P+1/2 pV^2+pgh_(loss,turns)+pgh_(loss,obstructions)+pgh_(loss,wall shear)=P_0

The final result is an equation that is useful but repulsively oversized, and it is certainly not well-established in theory.

One recent example of such phenomena is the Schrödinger equation that explains the behavior of matter at the atomic level. This relation just happens to work, and its derivation is intuitive rather than rational. It also has the simplest mathematical form among its possible competitors in terms of expressing nature.

We will now look at another example of such axiom-like relations that we usually ignore, although it is frequently encountered in our everyday life. Before revealing it as fully as we can, let us relate one situation where this effect is very apparent.

We usually move objects by pushing or pulling them. Suppose now we are on a motorboat and we have run out of gas in a place very close to the shore. We (the strong crew members) surely do not want to be carried away from the shore by the backwash from the waves. One of us has the brilliant idea to push on the sides of the boat until we reach harbor. What would you suggest? Some of us think that it is not a good idea because the boat is very heavy and our pushing will be negligible. It is true that the boat will not move. However, the failure has nothing to do with the weight of the boat. On the other hand, some other crew members suggest using oars, which will obviously work, but why? (Personally, with all my respect to other opinions, I would suggest using the phone to call the beach police to get some help; but this would distract us from our subject matter.)

Impulse, direction, and momentum

If you think about the “why” question above, you will guess that we are talking about impulse in the loose meaning of the word. In physics, impulse has a more precise definition. This definition arose from the need to describe an object’s ability to have an impact on other objects, but the idea is still vague: How do we quantify this ability in order to put some flesh on this notion? Let us try to figure out an answer to this question.

Now, let us consider a few possible ways of defining impulse that look reasonable. We may decide intuitively that an impact should be related to an object’s speed: the higher the speed the greater the impact. If you ever played marbles in your childhood, you will recall that the easiest way to dislodge the marbles in the targeted row is to cast your own marble as fast as you can. Impulse should also have a relation to mass. Certainly, the impact of as many as a thousand bullets aimed at a train will not move the train even a meter. These are some simple observations anyone can experience or have a feeling of from their daily life.

We also expect that this strange quantity should somehow be transferred by the interaction of two objects. One object colliding with another stationary object transfers something that causes the latter to travel in a direction. With this example, another important feature of our impulse idea emerges: direction. Those who like to play the game of American pool or billiards know this very well. (I am sure everyone does it for the noble reason to experiment the laws of physics.) It makes a significant difference in a collision of two masses if they hit each other at an angle.

Wait a minute! We have been talking about the effect of an object’s impact, but the object has something that it is carrying even before the impact, and this “something” is the reason why we have an impact in the first place. So, what is this “something”? Let us call it momentum so as not to violate the traditions of physics.

All this stuff so far is good, but we are not done yet: how should these ideas appear in our equations? Now, let us bring together all our findings. We know momentum manifests itself as the impact (P) of one object on another. From its effect (impulse), we understand that momentum is related to the mass (M) of the object and its velocity (V). We also know that momentum has a directionality, which is termed vectorial. Then, perhaps momentum is something like:

P = a x M + b x V

where a and b are constants. This seems acceptable since it satisfies our observation: the more the mass, the more the momentum. But for a stationary object (V=0), there is no point in talking about impulse; so the axM terms looks unnecessary. If there is no good reason for a physical quantity to appear in a physical equation, then the simplicity principle says it should be removed. Therefore, we look for a simpler alternative relation with only one term like below:

P = c x M2 x V5

where c is a constant. But this one is a highly non-linear relation with exponential terms, so it is really not looking good. Another problem with this equation is that it does not fit our daily experience very well. If we reconsider our train example, with a high velocity power term like this, even the very small bullets can have a considerable effect on a train, enough to move it in fact. As a simple example, let c be equal to 1, take 0.1 kg as the mass of a bullet and 105 kg (100 tons) as the mass of the train, and give 400 m/s velocity to the bullet. Assuming that the impulse of the bullet is transferred to the train (conservation of momentum), we roughly get:

Pbullet = c x m2 x Vb5 = 1011

Ptrain = c x M2 x Vt5 = 1010 Vt5

By equating both sides we roughly get, 1.6 m/s (5.7 km/h) for the train velocity.

A single bullet moving a big train at such speed? This is a very counter-intuitive result. However, we will not give up easily. How about if we try an expression which is more familiar?

P = d x MV2

where d is a constant. This relation also agrees with our intuition (i.e. it has mass and velocity terms proportional to P.) I can already hear some objections from those who are acquainted with physics saying “No! This is the energy formula of a body with mass M and velocity V.” Indeed, this equation is reserved for energy which is a non-vectorial quantity. As a matter of fact, none of the above is a correct description for momentum. The actual expression is, interestingly, the simplest of all possibilities:

P = MxV

So, why not the more complex ones but this, the simplest one? The rigorous answer is subtle and requires a thorough analysis of linearity and homogeneity of space, which could be the subject of another essay. But we repeat the remark that we made in the beginning: if there is a simpler and more beautiful way of describing a natural law, then that description often turns out to be the correct one in the end. In fact, for some giants of physics like Paul Dirac, the beauty of a theory is more important than its results and is a better indication of the theory’s correctness. There is more to it than that. There are whole theories like Dimensional Analysis which are implicitly based on the idea of writing down the equations in the simplest (and most beautiful) form.

So, the final point is that, like other fundamental relations in physics, the idea of impact or momentum is best described in the simplest and most intuitive way: P= MxV. And behold! This gives us exactly the relation that has passed all the scientific tests in the range of classical physics. This result implies that the beauties we see in nature can be explained in terms of the simplest possible physical relation. Pondering all these, one cannot help but ask how in the world a mindless, blind natural law could exhibit beauty based on simplicity.