It's good that it's divisible by 2
From time to time I patch my fellow physicists by saying that physics itself is too difficult for them. Modern physics has become more mathematical by 90%, if not 100%. It is common for physics teachers to complain that they cannot teach well because they do not have the appropriate mathematical apparatus at school. But I think that most often ... they simply cannot teach, so they say that they must have the appropriate concepts and mathematical techniques, especially differential calculus. It is true that only after mathematizing a question can we fully understand it. The word "compute" has a common theme with the word "face". Show your face = be calculated.
We were sitting with a colleague, Polish philologist and sociologist Andrzej, by the beautiful lake Mauda, Suwałki. July was cold this year. I don’t remember why I told a well-known joke about a motorcyclist who lost control, crashed into a tree, but survived. In the ambulance, he raved, “it’s good that he shared at least two.” The doctor woke him up and asked what was going on, what to divide or not to divide by two. The answer was: mv2.
Andrzej laughed for a long time, but then timidly asked what mv2 was about. i explained it E = mv2/2 this is the formula for kinetic energyquite obvious if you know integral calculus but don't understand it. A few days later he asked for an explanation in a letter so that it would reach him, a Polish teacher. Just in case, I said that there are no royal roads in Russia (as Aristotle said to his royal disciple Alexander the Great). They all have to suffer the same way. Oh, is it true? After all, an experienced mountain guide will guide the client along the simplest path.
mv2 for Dummies
Andrey. I would be dissatisfied if the following text seemed too difficult for you. My task is to explain to you what this clip is about.2. Specifically why a square and why we divide by two.
You see, mv is momentum, and energy is the integral of momentum. Simple?
For a physicist to answer you. And I ... But just in case, as a preface, a reminder of the old days. We were taught this in elementary grades (there was no middle school yet).
Two quantities are directly proportional if, as one increases or decreases, the other increases or decreases, always in the same proportion.
For example:
X 1 2 3 4 5 6 7 8 9
I 5 10 15 20 25 30 35 40 45
In this case, Y is always five times larger than X. We say that proportionality factor is 5. The formula describing this ratio is y = 5x. We can draw a straight line graph y = 5x (1). The proportional graph of a straight line is a uniformly ascending straight line. Equal increments of one variable correspond to equal increments of the other. Therefore, a more mathematical name for such a relationship is: linear dependence. But we are not going to use it.
1. Graph of the function y = 5x (other scales along the axes)
Let's turn now to energy. What is energy? We agree that this is some kind of hidden power. “I don’t have the energy to clean” is almost the same as “I don’t have the energy to clean.” Energy is a hidden force that lies dormant in us and even in things, and it is good to tame it so that it serves us, and does not cause destruction. We get energy, for example, by charging batteries.
How to measure energy? It's simple: a measure of the work he can do for us. In what units do we measure energy? Just like work. But for the purposes of this article, we will measure it in ... meters. How so?! We'll see.
An object suspended at a height h above the horizon has potential energy. This energy will be released when we cut the thread on which the body hangs. Then he will fall and do some work, even if he just makes a hole in the ground. When our object flies, it has kinetic energy, the energy of the movement itself.
We can easily agree that the potential energy is proportional to the height h. Carrying a load to a height of 2 hours will tire us out twice as much as lifting to a height h. When the elevator takes us to the fifteenth floor, it will consume three times as much electricity as on the fifth ... (after writing this sentence, I realized that this is not true, because the elevator, in addition to people, also carries its own weight, and considerable - to save the example, you have to replace the elevator, for example, with a construction crane). The same applies to the proportionality of potential energy to body mass. Transporting 20 tons to a height of 10 m requires twice as much electricity as 10 tons to 10 m. This can be expressed by the formula E ~ mh, where the tilde (i.e., the ~ sign) is a proportional sign. Double the mass and double the height equals four times the potential energy.
Giving the body potential energy by lifting to a certain height would not have taken place if it were not for gravity. It is thanks to her that all bodies fall to the ground (to the Earth). This force works so that the bodies receive constant acceleration. What does "constant acceleration" mean? This means that a falling body steadily and steadily increases its speed - just like a car starting off. It moves faster and faster, but accelerates at a constant speed. We will soon see this with an example.
Let me remind you that we denote the acceleration of free fall through g. It's about 10 m/s2. Again, you may be wondering: what is this strange unit - the square of a second? However, it should be understood differently: every second the speed of a falling body increases by 10 m per second. If at some point it moves at a speed of 25 m/s, then after a second it has a speed of 35 (m/s). It is also clear that here we mean a body that is not overly concerned with air resistance.
Now we need to solve an arithmetic problem. Consider the body just described, which at one moment has a speed of 25 m / s, and after a second 35. How far will it travel in this second? The problem is that the speed is variable and an integral is needed for correct calculations. However, it will confirm what we feel intuitively: the result will be the same as for a body moving uniformly at an average speed: (25 + 35)/2 = 30 m/sec. - and therefore 30 m.
Let's move to another planet for a moment, with a different acceleration, say 2g. It is clear that there we gain potential energy twice as fast - by raising the body to a height twice as low. Thus, the energy is proportional to the acceleration on the planet. As a model, we take the acceleration of free fall. And therefore we do not know a civilization living on a planet with a different force of attraction. This brings us to the potential energy formula: E = gmch.
Now let's cut the thread on which we hung a stone of mass m at a height h. The stone falls. When it hits the ground, it will do its job - it's an engineering question, how to use it to our advantage.
Let's draw a graph: a body of mass m falls down (those who reproach me for the phrase that it cannot fall up, I will answer that they are right, and therefore I wrote that it was down!). There will be a marking conflict: the letter m will mean both meters and mass. But we'll figure out when. Now let's look at the graph below and comment on it.
Some will think it's just clever numbering tricks. But let's check: if the body takes off at a speed of 50 km / h, it will reach a height of 125 m - that is, at the point where it stops for an infinitely short moment, it will have a potential energy of 1250 m, and this is also mV2/ 2. If we launched the body at 40 km / h, then it would fly at 80 m, again mv2/ 2. Now we probably have no doubt that this is not a coincidence. We found one of Newton's laws of motion! It was only necessary to set up a thought experiment (oh, sorry, first determine the acceleration of free fall g - according to legend, Galileo did this when dropping objects from the tower in Pisa, even then a curve) and most importantly: to have numerical intuition. Believe that the good Lord God created the world by following the laws (which he may have invented himself). Maybe he thought to himself, "Oh, I'll make laws so that they can be divided by two." That's a half, most of the physical constants are so incredibly strange that you can suspect the Creator of a sense of humor. This also applies to mathematics, but not about it today.
About a dozen years ago, in the Tatras, climbers called for help from one of the walls of Morskie Oko. It was February, cold, short days, bad weather. Rescuers got to them only at noon the next day. The climbers are already cold, hungry, exhausted. The rescuer handed the first of them a thermos of hot tea. "With sugar?" the climber asked in a barely audible voice. "Yes, with sugar, vitamins and a circulatory booster." “Thank you, I don’t drink with sugar!” - answered the climber and lost consciousness. Probably, our motorcyclist also showed a similar, appropriate sense of humor. But the joke would have been deeper if he had sighed, let's say: "Oh, if not for this square!".
For what the formula says, the relation E = mv2/ 2? What causes "square"? What is the peculiarity of "square" relations? That, for example, doubling the cause produces a fourfold increase in the effect; three times - nine times, four times - sixteen times. The energy we have when moving at 20 km/h is four times lower than at 40, and sixteen times less than at 80! And in general, imagine the consequences of a collision at a speed of 20 km / h. with the aftermath of a 80 km/h collision. Without any template, you can see that it is much, much larger. The ratio of effects increases in direct relation to speed, and dividing by two softens this up a bit.
* * *
The holidays are over. I have been writing articles for several years now. Now… I have no strength. I would have to write about the education reform, which also has good sides, but the decision was made on a non-subject basis by people who were suitable for what I am for ballet (I am significantly overweight and I am over 70 years old).
However, as if on duty, I will refer to another manifestation of elementary ignorance among journalists. Admittedly, nothing compares to a journalist from Olsztyn who devoted a long article to the issue of consumer fraud by manufacturers. Well, the journalist wrote, fat content was indicated on a pack of butter as a percentage, but it was not explained whether it was per kilogram or per whole cube ...
An inaccuracy written by journalist A.B. (fictitious initials) in Tygodnik Powszechny of July 30 this year, thinner. He stated that, according to a CBOS study, 48% of people who consider themselves very religious take a certain X attitude (no matter what it is, it doesn't matter), and 41% of those who participate in religious practices several times a week support X. This means, the author writes, that more than two-fifths of the most active Catholics do not recognize X. I tried for a long time to find out where the author got these two-fifths, and ... I don’t understand. There is no formal error, since indeed, mathematically speaking, more than two-fifths of the respondents are against X. You can simply say that more than half (100 - 48 = 52).
