## Tuesday, January 11, 2011

### Mathematics in Physics - Distance, Time, Speed

We are used to use mathematics with physical concepts quite automatically, without thinking what is going on in a formula.

Let’s clarify a bit what is going on under the imaginative hood in a physics formula.

Look at the speed. Everyone knows that speed is quotient of distance and time. You obtain distance, then divide it by time. But, let’s take a close look at these two physical concepts separately.

Distance. We have a physical feeling what a distance is. We can measure it. We can count steps, kilometres. Now look at our counts that comes from distance. Let’s say we counted 5 meters. No matter what we have counted we have a number 5. That’s the count. Can be a count of meters, or apples, or atoms. That’s how mathematics enters physics. However, once we are in physics, suddenly it matters WHAT is counted! While mathematics does NOT CARE when you add 3 + 2 to obtain 5 where did you get those 2 and 3, in physics it matters what you count. In physics it matters where the counts come from!

Let’s consider example of 5 baskets and each basket has 3 apples. How many apples are there? Of course you will multiply 5 baskets by 3 apples to get 15 apples. But, one may ask how I can multiply baskets and apples. These are two different things. The trick here is you first abstract, or, obtain separately, or extract, a COUNT of baskets, keeping in mind WHAT you counted! That’s the count (or number!) five. Then, you obtain a COUNT of apples, which is 3. These COUNTS are now separated from objects that generated the counts, form what was counted! And since they are PURE counts, we can do with them what we can do with counts (numbers) in math! We can add them, subtract them, multiply, or divide. In this case will multiply COUNTS (not apples, not baskets). Let’s do it. 5 x 3 = 15.

In physics, it is custom not to use parenthesis but just to put a letter beside the number to label what is counted. Hence, for meters it will be 5m, for time it will be 5sec, etc. And it is up to you what are you going to do with these counts! You can divide them, add, multiply etc. What are you going to do, and why, has nothing to do with math! It’s your intuition or physical knowledge, or inspiration! Math is like an obedient servant who will do what you told him to do! Add 3 + 2! Sure, 3 + 2 = 5. But, this “servant” does not know what you have counted, what you have added and, moreover, what would be the reason for addition!

Let’s look at our speed example! You measured distance, and obtained, say 10 meters. Then, simultaneously, you measured, counted time, and obtained, say, 5 sec. Now, inside the World#1, i.e. the world of distance, time, our physical real world around us, only WE can decide what makes sense to do with measurements and counts. We can easily say let’s multiply 10 by 5 and obtain 50. But, that really does not help us if we want to get SPEED. To get speed, we need to divide 10 by 5 to get how many meters per second we have been traveling! So, let’s do it. Dividing it we obtain 10 / 5 = 2. Now, what is two? Is it distance? No! Is it time? No! Ok, you can say it is speed, but we do not know that yet. We dealt so far with distance and time. You can say that 2 we have obtained is actually 2 meters per second, and name it as speed. Look how we have dragged the description what is counted to the new physical concept, speed! And, moreover, we can quantify speed using the counts of distance and time. Don’t forget, number two is still only a number, but, you drag the label with it, the description what you counted or even more how you obtained it. It is 2 meters per second or 2 m/s. That unit designation [m/s] i.e. [meters/second] tells you the information form World # 1. It tells you that you have counted, first, distance, in meters, than you divided that COUNT with the count obtained from counting time. What you did is you have divided COUNT of meters by COUNT of seconds, and not METERS by SECONDS.

So, we do have these straightforward steps, COUNT of meters / COUNT of seconds to somewhat abbreviated notation, COUNT(m) / COUNT(sec) to very abbreviated notation 10m / 5sec. You can see that units in physics are nothing more than labels to COUNTABLE objects that we have a special interest or motivation to count!

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