The Value of Mechanics

Mechanics is an exact science. Well, it's an exact science based off of an approximation (but if you account for relativity I suppose it does become exact again...)

The value of this is that you can predict with very high precision what will happen in an experiment. To a large extent, you can get by without any experimental data whatsoever- given conservation of energy and mementum (and a few other 'first principles') most of classical mechanics is derivable. On the other hand, it's not too difficult to think up an experiment to test many of the ideas that come out of mechanics.

Pretty much the simplest mechanics situation is linear, constant acceleration kinematics. This is a perfect example of what makes mechanics the science that it is: you start off with a large amount of real world data which is messy and unorganised. From this, we realise that there is one easy to measure quanitity for a moving object: its position. From this we can define the derivatives of its position with respect to time. These are called velocity, acceleration, jerk, snap, crackle, pop (yes, really) and so on. Or, if you want to be boring, they are called the first, second etc. time derivatives of displacement.

The amaxing thing here is that we have not yet said anything, just realised from our everyday understanding of the world that position, velocity and acceleration are going to be important. Now, let's see what happens if we say that acceleration is constant...

... well I can't really show you too well here. But what happens is that because we can link acceleration to velocity and thus to distance, setting it to a constant immediately gives us a set of equations which define a particles movement entirely. That is what kinematics is all about. And the science of mechanics is largely about that "Aha!" moment where suddenly, everything is determined and only one motion is possible. But science is not just about theory, and doesn't taking an infinite sum of infinitely many infinitesimal changes in velocity just seem like asking for trouble? What if the maths does not work when applied to real life?

Well that's why, ultimately, experiment is the final arbiter. And usefully, position is in fact very easy to measure. As is time. Velocity and acceleration are a little harder seeing as using our derived equations would rather be cheating, seeing as those are what we're trying to test! However, since position is really the only thing that matters and is directly measurable, that's all we need to measure- the other properties can be treated as theoretical for all we care.*

So, let's take a simple equation and see what it predicts: S=ut+1/2at^2. How high a tower dowe need to drop a ball from for it to take exactly one second to land? Well, S=(0)(1)+1/2(9.81)(1^2)

S=0+4.905m

We can test that. We can also do tons of other easy experiments involving dropping items. Our formula suggests that (even if we don't know the value of g) the time taken to fall will increase with the square root of the height. We can give the particles an initial push and test what effect that has. Mechanics very often will have nice, easy ways to test an equation. The motion of falling particles is a very easy way to test the equations of motion, because acceleration is always constant. And in case you don't like dropping obects, smooth inclined planes offer another way of making measurements and another way to take multiple readings by varying the angle.

So mechanics is heavily theoretical. But it is in many paces easily tested by experiment. This made it a source of many easy and some not-so easy problems while classical mechanics was being formulated. But now we have relativity and quantum mechanics, so surely we do not need it any more? Classical mechanics helps see the world in a certain analytical light. The step-by-step procedure of "observe, generalise, define, calculate, experiment" is very intuitive. We noticed that motion seemed to follow patterns. We invented concepts of velocity and acceleration to help us define position as a function of time. We defined a series of time derivatives, and then worked out the mathematical consequences. Finally, experiments clearly show that these rules are very accurate and broadly applicable. The abstract chain of reasoning comes home to the harsh light of observation, and emerges unscathed. That is the value of mechanics, which is the most well-known area of physics and certainly one of the most emphasised.

*When I say "theoretical construct" I do not mean to say that velocities and accelerations do not exist. Feynman once intended to ask a group of philosophers whether the 'inside of a brick' was an essential object (he wanted to know whether the phrase essentail object meant theoretical). Whenever you cut open the brick, you only ever see the new surface, not the inside. But it makes life that litle bit simpler to assume that there is an inside that accounts for the brick's density and the constantly appearing 'new surfaces' that pop into existance when you cut a brick in half. He never got to ask the question though, however, because he led with "Is a brick an essential object?" and the room erupted into philosophical chaos.