Before we dive any further into Newton, forces and other things, lets go a bit into the fantastic world of vectors.
first, a definition. Vectors are defined as things that possess a magnitude as well as direction, in contrast to lowly scalars, which only get magnitude.
"So what is the big deal?", I am sure you ask. Well, it does not seem like much, but let me try to explain it like this.
If you wanted to lift a rock, pushing down on it would not lift it. Instead , you would have to probably pull the rock up, and there you go, one rock lifted. Here, the direction of the force is just as important as the magnitude. So, we can then say that force is a vector. Now how about travelling. Let us say that I have taken my car, driven down from Bangalore all the way to Chennai, and back again. So I have covered 600 (or so) kilometres doing this. My tyres and my petrol tank make that clear. But the fact of the matter is that I have not really moved 600 kilometres from my home...I am back there, lounging. So while the distance travelled might have been 600km, my displacement from my initial condition was zero.
So for now, just remember that a vector is rather more complicated than a scalar, and although the vector may be very strong (has high magnitude), it may not do much to you (beacuase the direction of th high magnitude is low.
We wont even go into tensors, and where they fit into the situation. We can worry about tensors we we come face to face with one!
Friday, July 6, 2007
Zeno's Paradox
I must say, I did intend to complete a discussion of Accelaration and Newtons laws today, but in trying to figure out the principle of differentiation, I came across this pretty interesting one which is a sort of introduction to limits and infinity (and beyond too!).
I promise that this will help while we discuss differentiation.
This one is about the greeks. There was this philosopher named Zeno, who was generally regarded as hot stuff in the philosophical circles of Greece. What did he come up with? Well, mainly he was of the school of thought that said that all motion is impossible, or some such thing. In order to prove it, he came up with these paradoxes, which at first sight seem terribly convincing. For example, there is his famous one about Achilles and the tortoise. (For those who came in late, Achilles was a mighty greek hero. He was a great runner. And showed a clean pair of heels to all his competitors. And was shot in the heel, causing his death. For the unabridged version, read the Illiad, by Homer). Below is a quote from the Wiki (always useful).
In the paradox of Achilles and the tortoise, we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is so fast a runner, Achilles graciously allows the tortoise a head start of a hundred feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run a hundred feet, bringing him to the tortoise's starting point; during this time, the tortoise has "run" a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, during which the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, Zeno says, swift Achilles can never overtake the tortoise. Thus, while common sense and common experience would hold that one runner can catch another, according to the above argument, he cannot; this is the paradox.
Now, try to spot the hole in that logic. Its actually quite tough. But I shall spare you, the reader from thinking this through. The problem with Zeno's logic is this. He assumes that for there are an infinite number of steps for Achilles to take to overtake the tortoise (Infinitely small steps). And since it takes a finite time to complete each step, it will take infinite time to complete infinite steps. In functions, this is called a one-to-one correspondence. But this is not necessarily true. Let us quantify the tortoise and Achilles problem like this.
In the case of Achilles and the tortoise, suppose that the tortoise runs at a constant speed of v metres per second (ms-1) and gets a head start of distance d metres (m), and that Achilles runs at constant speed xv ms-1 with x > 1. It takes Achilles time d/xv seconds (s) to reach the point where the tortoise started, at which time the tortoise has travelled d/x m. After further time d/x2v s, Achilles has another d/x m, and so on. Thus, the time taken for Achilles to catch up is
The above value is a finite number, which means that Achilles will eventually catch up with the tortoise. Notice, that without a definition that implies that distance and time can be related, this problem cannot be solved at all.
p.s: Almost all the material is lifted from the Wiki. The link is given below for those who wish to have a full explanation (and to get thoroughly confused...I was).
here
I promise that this will help while we discuss differentiation.
This one is about the greeks. There was this philosopher named Zeno, who was generally regarded as hot stuff in the philosophical circles of Greece. What did he come up with? Well, mainly he was of the school of thought that said that all motion is impossible, or some such thing. In order to prove it, he came up with these paradoxes, which at first sight seem terribly convincing. For example, there is his famous one about Achilles and the tortoise. (For those who came in late, Achilles was a mighty greek hero. He was a great runner. And showed a clean pair of heels to all his competitors. And was shot in the heel, causing his death. For the unabridged version, read the Illiad, by Homer). Below is a quote from the Wiki (always useful).
In the paradox of Achilles and the tortoise, we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is so fast a runner, Achilles graciously allows the tortoise a head start of a hundred feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run a hundred feet, bringing him to the tortoise's starting point; during this time, the tortoise has "run" a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, during which the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, Zeno says, swift Achilles can never overtake the tortoise. Thus, while common sense and common experience would hold that one runner can catch another, according to the above argument, he cannot; this is the paradox.
Now, try to spot the hole in that logic. Its actually quite tough. But I shall spare you, the reader from thinking this through. The problem with Zeno's logic is this. He assumes that for there are an infinite number of steps for Achilles to take to overtake the tortoise (Infinitely small steps). And since it takes a finite time to complete each step, it will take infinite time to complete infinite steps. In functions, this is called a one-to-one correspondence. But this is not necessarily true. Let us quantify the tortoise and Achilles problem like this.
In the case of Achilles and the tortoise, suppose that the tortoise runs at a constant speed of v metres per second (ms-1) and gets a head start of distance d metres (m), and that Achilles runs at constant speed xv ms-1 with x > 1. It takes Achilles time d/xv seconds (s) to reach the point where the tortoise started, at which time the tortoise has travelled d/x m. After further time d/x2v s, Achilles has another d/x m, and so on. Thus, the time taken for Achilles to catch up is
The above value is a finite number, which means that Achilles will eventually catch up with the tortoise. Notice, that without a definition that implies that distance and time can be related, this problem cannot be solved at all.
p.s: Almost all the material is lifted from the Wiki. The link is given below for those who wish to have a full explanation (and to get thoroughly confused...I was).
here
Subscribe to:
Posts (Atom)