Chapter 5 Chapter 4 Learn Algebra

Have a good night.To be honest, the word night is not used properly enough. The relation of the projectile to the sun has remained constant.In astronomical terms, the base of the projectile is day and the top is night.Therefore, the night and day mentioned in the narrative of this book refer to the time between rising and setting of the sun on earth. The main reason why the three travelers slept so peacefully was because, although the projectile was moving with great speed, it seemed absolutely still inside.No movement is sufficient to suggest that it is flying through space.When a body is in a vacuum, or is moving with it in the surrounding air, no matter how great the speed is, it cannot affect the human organism.The ground moves 90,000 kilometers per hour, but who noticed its speed?In this case the sensations caused by motion and rest are exactly the same, and therefore motion has no influence on the whole body.An object at rest will remain at rest forever if it is not pushed by an external force.Likewise, an object in motion will remain in motion forever unless it encounters an obstacle.The invariance of motion or rest is called inertia.

It was therefore natural for Barbicane and his companions to think that they were in a state of absolute rest, being confined within the projectile.Moreover, even if they remained outside the projectile, the result would still be the same.If the moon before them had not grown larger, and the earth beneath them had not grown smaller, they might have sworn that they were at a state of absolute stagnation. On the morning of December 3, they were awakened by an unexpected cry of joy.There was the crowing of a rooster in the carriage. Michelle.Adam was the first to get up. He climbed up to the vault and covered a half-open wooden box.

don't bark!he whispered.This idiot almost ruined my event! At this time, Nicholl and Barbicane also woke up. Where did the cock come from?Nicholl asked. No!My friends, Michel hastened to answer, I am urging you to wake up with country songs! Having said this, he suddenly let out a loud clucking rooster cry.Even the proudest quail would be proud of it. The two Americans couldn't help laughing. Good job, said Nicholl, looking suspiciously at his companion. Yes, Michelle replied, it was always a joke back home.Very Gallic.In high society, we also learn to crow like this! Then, to change the subject, he said to Barbicane: Do you know what I have been thinking all night?

No idea, replied the club president. I am thinking of our friends at the Cambridge Observatory.You have of course noticed that I am ignorant of mathematics, and I could never have guessed how the scientists at the Observatory calculated the initial velocity the projectile must have to leave the Columbia to reach the Moon. You mean, Barbicane corrects him, the velocity to the line of weightlessness where the gravitational forces of the earth and the moon are balanced, because there, that is to say, about nine-tenths of the way the projectile travels, it will be due to Its own weight falls to the moon.

Even so, Michelle replied, but, let me repeat, how is the muzzle velocity calculated? Nothing could be more convenient, said Barbicane. Can you count?Michelle.Adam asked. Of course.Nicholl and I could both have calculated it ourselves if the data from the observatory hadn't saved us the trouble. Well, my old Barbicane, answered Michel, even if I were to be cut in half from head to toe, I would not be able to solve the problem! That's because you don't know algebra, answered Barbicane quietly. ah!You put it nicely, you experts in X always think that all you have to do is say algebra and everything will be solved.

Michel, said Barbicane, do you believe that iron can be struck without a hammer, and the field can be plowed without a plow? That would be too difficult. Well, algebra is a tool, like a plow or a hammer, and a good tool for those who know how to use it. Really? True. Can you wave this tool in my face? If you like. Show me how to calculate the muzzle velocity of our carriage? Yes, my dear friend.I can deduce absolutely correctly the initial velocity of the projectile from all the data in this problem, that is, from the distance between the center of the earth and the center of the moon, the radius of the earth and the moon, the mass of the earth and the moon, and just list a simple The formula will do.

Let's look at your formula. You will see in a moment.However, I will not draw you the curve that the cannonball actually passes between the moon and the earth, because these two celestial bodies are also orbiting the sun.Yes, we assume that the two celestial bodies are stationary, and that will suffice. Why? Because it is enough to find the answer to the so-called three-object problem, and calculus is not the most advanced method for solving this problem. So, Michelle.Adam said in a teasing voice, mathematics can't solve the problem? Of course not, replied Barbicane. Well, maybe the lunar calculus is more advanced than yours!Also, by the way, what is calculus?

This is a method of calculation that is the exact opposite of differential calculus, Barbicane replied solemnly. Thanks. In other words, we can differentiate a finite amount of numbers. At least this sentence was clear and understandable, Michel replied with an air that could not be more satisfied. Now, continued Barbicane, with a piece of paper and a pencil, I hope to be able to produce the formula you require in half an hour. At this point, Barbicane concentrated on his work, Nicholl continued to observe space, and their companions also took this opportunity to prepare breakfast. Before half an hour had passed, Barbicane raised his head and handed Michel a page full of mathematical symbols.Ah, look, there's a general formula in the middle:

(Attachment 1) What does it mean?Michelle asked. The formula means, Nicholl replied, that one-half multiplied by the difference between the square of v and the square of v0 is equal to gr multiplied by square brackets x divided by r minus 1 plus m prime multiplied by parentheses d and x The difference between r minus d and the difference between r parentheses and square brackets x rides y, y rides z, z climbs p's back, michelle.Adam laughed.Can you read this thing, Captain? It doesn't get any clearer than that. What?Michele said, nothing could be clearer than that, and I dare not take it anymore.

You can play tricks, Barbicane retorted.You said you were going to learn some algebra, but now you're bored again! I would rather let someone hang me up! In fact, Nicholl studied Barbicane's formula with an expert eye, and he said, I think your formula is very good, Barbicane.This is a complete formula for strength in these kinds of sports, and I have no doubt that it will give us the answers we are looking for! I wish I could read it!Michelle said loudly, even if it cost Nicholl ten years of life, I would be willing! Listen, then, continued Barbicane.One half multiplied by the difference between v squared and v0 squared, this formula tells us that this is one half of the change in kinetic energy.

Very good, does Nicholl know what this means? No doubt, Michel, replied the captain.All these symbols, which you find mysterious and incomprehensible, are the clearest, most intelligible, and most logical language to those who can read. Do you mean, Nicholl, asks Michel, that you must be able to find the initial velocity that a projectile must have through these hieroglyphs, which are more difficult to understand than the script of the Egyptian spirit bird? There is no need to doubt, replied Nicholl, and I might even say that I can tell you the velocity of a projectile passing any point. can you swear I swear. That means you're as smart as our club president? No, Michelle.The most difficult was the work done by Barbicane.Because such an equation is listed, all conditions of all aspects of the problem must be considered.All that remains is a matter of arithmetic operations, by applying the four rules of arithmetic. That's so beautiful!Michelle.Ah Dang replied that he had never done addition correctly once in his life, so he said that addition is like a Chinese jigsaw puzzle, and many different answers can be obtained. At this moment, Barbicane said to Nicholl that if Nicholl thought about it a little, he would be able to formulate this formula. I don’t know, Nicholl said, because of your formula, the more I think about it, the more magical it becomes. Now, listen carefully, said Barbicane to his lay companion, for you will soon see that all these signs have their meaning. All ears, Michelle said with a helpless look. d is the distance between the center of the earth and the center of the moon, Barbicane said, because gravity must be calculated from the center. I understand this. r is the radius of the earth. r, radius.I agree. m is the mass of the earth; m is the mass of the moon.In fact, we must consider the mass of two objects that are attracted to each other, because the gravitational force is proportional to the mass. of course. g stands for gravity, which represents the distance an object travels in one second falling toward the earth.do you understand? So clear!Michelle replied. Now I shall denote by x the constantly changing distance of the projectile from the center of the earth, and by v the velocity of the projectile at this distance. very good. Finally, v0, which appears in the equation, represents the velocity of the shell after it has passed through the atmosphere. In fact, says Nicholl, the velocity must also be calculated at this point, since we already know that the initial velocity is exactly one and a half times the velocity after passing through the atmosphere. I can't figure it out here again!Michelle said. But the question is very simple, said Barbicane. But for me, it's not that simple, Michelle replied. That is to say, when the projectile rises to the final limit of the earth's atmosphere, it has lost one-third of its initial velocity. To lose so much? Yes, my friend, it is only because of the friction of the atmosphere.You naturally understand that the faster it goes, the greater the resistance of the air. This, I agree, replied Michelle, I can understand it too, it's just that the sum of your v squared and v0 squared is banging around in my head like a nail in my pocket! This is the first item in the algebra, continued Barbicane.To solve this problem for you, we substitute in known numbers, that is, we substitute in values ​​we already know. You should solve me!Michelle replied. Some of these signs are known, said Barbicane, and the rest can be deduced. I'll do the math, Nicholl said. Let us now turn to r, continued Barbicane. r is the radius of the earth, that is to say, the radius of the earth at the latitude of Florida, our starting point, equal to 6.36 million meters. d is the distance between the center of the earth and the center of the moon, equal to fifty-six radii of the earth, that is to say Nicholl calculated quickly. That is, he said, when the moon is at perigee, that is, at its closest point to the earth, it is equal to 356.72 million meters. Very well, said Barbicane.Now, that is to say, the ratio of the mass of the moon to the mass of the earth is equal to one to eighty-one. Very good, said Michelle. g is gravity, and Florida's gravity is 9.81 meters.so y is equal to Six thousand two hundred and twenty-six thousand square meters, Nicholl replied. So what now?Michelle.Adam asked. Now, since these symbols have been substituted by numbers, Barbicane replied, I will now look for the data of v0, that is to say, the velocity of the projectile when it leaves the atmosphere and reaches the point where the gravitational forces of the earth and the moon cancel.Since the speed at this time is equal to zero, I can say that the point where the two gravitational forces are equal is at d, that is to say, at nine tenths of the distance between the centers of the two celestial bodies. I had a vague sense that it should, too, Michelle said. I can therefore also say: x equals nine tenths of d, v equals zero, and my formula then becomes Barbicane quickly wrote down his equation on paper: (Attachment 2) Nicholl glanced greedily. Exactly!Exactly!he said aloud. understand?said Barbicane. It's as clear as if written in flames!Nicholl replied. You two are so nice!Michelle muttered. Do you understand now?Barbicane asked him. Do I understand?Michelle.Ah Dang exclaimed, that is to say, my head exploded! Hence Barbicane again says that the square of v is equal to two gr multiplied by the parenthesis one minus ten r of nine d, minus one eighty-first, multiplied by ten r of d, minus r of the difference between d and r. . Now, Nicholl said, the speed of the cannonball after passing through the atmosphere can be found by simply performing calculations. So, as a mathematician who can solve all difficult problems skillfully, Nicholl began to calculate at a frightening speed.In just a moment, division and multiplication were lined up under his fingers in a long line.Numbers rolled across the white paper like hail.Barbicane followed him closely with both eyes, while MichelAdam held his head, which was beginning to ache, with both hands. How about it?asked Barbicane, after a few minutes' silence. very good!After calculation, Nicholl replied that the velocity of the projectile when it leaves the atmosphere and moves toward the place where the two gravitational forces are equal should be It should be Barbicane said. Eleven thousand and fifty-one meters. ah!Barbicane jumped up and said: What did you say? Eleven thousand and thirty-one meters. Damn it!There was a cry from the club president, who made a desperate gesture. What happened to you?Michelle.Adam asked in amazement. And ask me what's the matter!The current velocity has been reduced by one-third due to the friction of the air, so the initial velocity should be 16,576 meters!Nicholl replied. The Cambridge Observatory stated that a muzzle velocity of only 11,000 meters is sufficient.It is this speed that propels our shells away from the earth! How about it.Nicholl asked. How about it!This speed is not enough! ah? We can't reach the zero-gravity line! damned! We couldn't even cover half the distance! damn it!Michelle.Adam suddenly jumped up and cried, as if the projectile was about to hit the earth. We're going to re-land to Earth!