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Simple and Beautiful Momentum
May 1, 2008

Every occurrence in nature obeys some kind of relation that has been put in operation in the universe, and most scientists probably believe that humans have the skill to represent that relation to themselves mathematically to a certain extent. Many of the relations which are observed are accepted as independent facts until someone comes up with a method to derive them from more fundamental facts or relations. In this sense, the academic field of physics accepts some “axiom-like” relations that explain events well, but we cannot derive them from more fundamental relations or cannot question why they hold true. Another common property of such axiom-like relations is that they turn out to be the simplest of all the possible alternatives. This is the principle of simplicity, which is held by physicists to be such a deep and non-trivial feature of our universe that it indicates a preference for simplicity over complexity.

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.