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.

Once the multiplication is done, we have to go BACK to our world of baskets and apples, in the world that generated counts, in the world where, actually, matters what is counted. Note that transition. While in math world (let’s call it World #2) we deal only with counts when we return to the world of countable objects (apples, baskets, let’s call it World # 1) we do take care which counts belong to which object!! And that would be the major approach, trick to master the whole physics, or, any applied mathematics discipline (economics, finance, engineering, trading, biology). In all these disciplines you have a separate thinking, separate rules, logic, specific for that particular discipline, that defines some kind of relation between countable objects! The rules what and why is counted, in any of these disciplines, are quite alienated from mathematics. In most of the cases these rules have nothing to do with math! You do not need mathematics to come up with a concept of basket, apple, distance, time, atom, speed, force, cars, money, water, air. You do not need mathematics to say I will buy 12 litres of gas. And, say, later you decide to buy additional 5 litres of gas. Total would be 12 + 5 = 17. Mathematics enters only when you dealt with pure counts, and not WHY you decided to by first 12 then 5 litres! Math enters when you wanted to add these two numbers. For you, it matters that those numbers represent litres of gas, but from math point of view you could add the number of the buttons on your coat. These are all concepts from World#1 (physical world, economics world...) and each discipline has its own rules WHAT to count and WHY. Once you decide you want to quantify the relationships between objects in Labelling counts. In the World # 1 you will need only to take care, keep tab, what is counted and when, and then, somehow LABEL, those counts, most likely with an object name what is counted. For example, you can label count 3, in the basket-apples example, with apples to get 3(apples). In this case “apples” is the label WHAT is counted. Then, you can label number 5 with 5(baskets).

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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  1. I like this post because you really talk about some of the huge simplifications our notation has embedded in it. Students who say I'm good at math can still struggle in physics because of exactly what you're talking about here.

    It also makes me think about more complicated combined units. I was explaining the acceleration due to gravity to a student once and I said something like "you know, 9.8 meters per second . . . per second" and she said "oh! Now I get it". Lately I've switched to 22 mph/s which makes a lot more sense to students.

  2. Thank you, Andy, for the comment. It is an elegant method to use mph as a unit for speed and divide it by seconds to explain acceleration. Meters per second per second is way more confusing.

    Possibly less clear example, but that might clarify a bit definition, could be to say, for instance, we pass the distance of 1 meter in the first second. Then, in the second second we passed the distance of 2 meters (total 3 meters). The speed in first second was 1m/s. The speed in second second is 2m/s. The difference in speeds between these two seconds is 1 m/s. Then, let's say, in the third second, we passed the distance of 3 meters. The speed in the 3rd second is 3 m/s, but the speed DIFFERENCE between 3rd and 2nd second is still 1 m/s. Now, for each second we keep this constant difference in speed, and, because of that we can say that after, for example, 6 seconds the speed will be 6 m/s. So, each second the speed is increasing. For how much? For 1 m/s per each second. End, eventually this will be the point of enlightenment why we say that difference in speed, i.e. acceleration is said to be 1 m/s per second, or 1 (m/s)/s or terribly confusing 1 m/(s2) (squared) :-)

  3. Still thinking...haha. Perhaps, using the word "each" as in "for each second", things may be a bit more clear. Like, the speed change is 2 meters per second for each second. Or, acceleration is 9.81 m/s for each second the body is traveling in the gravitational field.