Chapter 5 Chapter 4 Learn Algebra

Spend the night safely.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.Thus, references to "night" and "day" in the narratives 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 the object is in a vacuum.Or when the surrounding air moves together with it, no matter how great the speed is, it cannot affect the human body.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 was not getting bigger, and the earth below them was not getting smaller, they might have sworn he said they were at an absolute standstill.

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. Michel Ardan was the first to get up, 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 you get the cock?" Nicholl asked. "No! My friends," replied Michel hastily, "I am urging you to wake up with country songs!" Speaking of this, he suddenly let out a loud "clack, cluck, cluck" 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," replied Michel, "there's always a joke like that in our country. It's very Gallic. We crow like that in high society!" Then, to change the subject, he said to Barbicane: "Do you know what I have been thinking about all evening?" "No," replied the club president. "I was thinking of our friends at the Cambridge Observatory. You have of course noticed that I know nothing of mathematics, and I could never have guessed how those scientists at the Observatory calculated the initial velocity the projectile must have to leave the Columbia to reach the Moon."

"You mean," Barbicane corrected him, "the velocity to the line of weightlessness where the gravitational forces of the earth and the moon are in equilibrium, since there) that is to say, about nine-tenths of the distance of the projectile's travel, it would fall to the moon by its own weight." "Even if that's the case," Michelle replied, "But, let me repeat, how is the muzzle velocity calculated?" "There is nothing more convenient," said Barbicane. "Can you count?" asked Michel Ardan. "Of course. Nicholl and I could have calculated it ourselves, if the materials at the observatory hadn't saved us the trouble."

"Very 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!" "It's because you don't know algebra," Barbicane replied quietly. "Ah! You put it so nicely. You experts in X always think that all you have to do is say algebra." "Michel," said Barbicane, "do you believe that the iron can be struck without a hammer, and the field can be plowed without a plow?" "That would be very difficult." "Well, algebra is a tool, like a plow or a hammer, and a fine tool for those who know how to use it."

"Really?" "It's absolutely true." "Can you wield this tool in front of me?" "If you like." "Show me how to calculate the muzzle velocity of our carriage?" "Yes, my dear friend. I can deduce the projectile absolutely correctly from all the data on the subject, 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. muzzle velocity, and only a simple formula is required.” "Let's see your formula." "You will see in a moment. However, I will not draw you the curve that the cannonball actually traverses between the moon and the earth, because these two celestial bodies also orbit the sun. Yes, we assume this It is enough that the two celestial bodies stand still."

"why?" "Because it is enough to find the answer to the so-called 'three-body problem', and calculus is not the most advanced method for solving this problem." "So," said Michel Ardan in a teasing voice, "mathematics can't solve the problem?" "Of course not," replied Barbicane. "Okay: Maybe the lunar people's calculus is more advanced than yours! Also, by the way, what is calculus?" "It's the exact opposite of differential calculus," replied Barbicane gravely. "thanks." "In other words, we can differentiate a finite amount of numbers."

"At least the sentence is clear and intelligible," replied Michel, with an air that could not have been more satisfied. "Now," continued Barbicane, "given a piece of paper and a pencil, I hope to be able, in half an hour, to produce the formula you require." 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 showed Michel Ardan a page full of mathematical symbols. There was a general formula in the middle: 1/2(v2-v02)=gr [r/x-1+m'/m(r/(dx)-r/(dr))] "What does this mean?..." Michelle asked.

"The formula means," replied Nicholl, "that one-half multiplied by the difference between the square of v and the square of v is equal to r minus one plus m of m multiplied by square brackets times the small Brackets The r of the difference between D and X minus the r of the difference between D and r Parentheses Square brackets..." "X rides y, y rides, z climbs p's back," laughed Michel Ardan. "Can you read this thing, Captain?" "It doesn't get any clearer than that." "What?" said Michelle. "It's never clearer than that, and I don't dare to learn it any more."

"You can play tricks," retorted Barbicane. "You said you were going to learn some algebra, but now you're bored!" "I'd rather have someone hang me up!" "As a matter of fact," said Nicholl, studying Barbicane's formula with an expert eye, "I think your formula is very good, Barbicane. It is a complete formula for power in these movements, and I No doubt it can give us the answers we're looking for!" "I wish I could understand it!" Michelle exclaimed. "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-square and v-zero-square, this formula tells us that this is half of the change in kinetic energy." "Very well, does Nicholl know what this means?" "Undoubtedly, Michel," replied the captain. "All these symbols that you think are mysterious and incomprehensible, to those who can read, they are the clearest, most intelligible, and most logical language." "You mean, Nicholl," asked 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 doubt," replied Nicholl, "and I may even say that I can tell you the velocity of a projectile passing any point." "Can you swear?" "I swear." "That means you are as smart as our club president?" "No, Michel. The most difficult thing is the work done by Barbicane. For such an equation must take into account all the conditions on all sides of the problem. All that remains is a matter of arithmetic operations, using only the four rules of arithmetic." The rules will do." "That's so beautiful!" Michel Ardan replied. He had never done addition correctly once in his life, so he said that addition "is like a Chinese jigsaw puzzle, with many different answers." 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 wonderful it becomes." "Now, listen carefully," said Barbicane to his lay companion, "you will soon see that all these symbols have their meaning." "All ears," said Michel with an air of resignation. "d is the distance between the center of the earth and the center of the moon," said Barbicane, "for gravity must be calculated from the center." "I understand that." "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, the distance an object travels in one second falling toward the earth. Got it?" "Too clear!" Michelle replied. "Now, I denote by X the changing distance of the projectile from the center of the earth, and by Y the velocity of the projectile at this distance." "very good." "Finally, the v zero that appears in the equation represents the velocity of the shell after it passes through the atmosphere." "In fact," said Nicholl, "the velocity at this point must also be calculated, 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 again these days!" Michel 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 due to the friction of the atmosphere. You understand, of course, that the greater the speed it goes, the greater the resistance of the air." "Well, I agree," Michele replied, "I can understand that too, it's just that your 'sum of V square and V zero square' is banging around in my head like a nail in my pocket!" "This is the first term in algebra," continued Barbicane. "In order to solve this problem for you, we substitute in known numbers, that is, we substitute in values ​​we already know." "You'd better get rid of me!" Michelle replied. "Some of these signs are known," said Barbicane, "and the rest can be deduced." "I'll crunch the numbers," Nicholl said. "Let us now turn to r," resumed 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, is equal to 6.36 million meters. d is the distance between the center of the earth and the center of the moon, which is equal to fifty-six earth radii, that is, Say……" Nicholl calculated quickly. "That is," said he, "when the moon is at perigee, that is, at its closest point to the earth, it is equal to 356,720,000 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 well," Michelle said. "g is gravity, and gravity in Florida is 9.81 meters. So y equals..." "Six thousand two hundred and twenty-six thousand square meters," Nicholl replied. "And now?" asked Michel Ardan. "Now, since these symbols have been substituted by numbers," replied Barbicane, "I will now look for the data of v zero, 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 each other out. Since At this time, the speed is equal to zero, and I can say that the point where the two gravitational forces are equal is on the mountain, that is to say, on nine-tenths of the distance between the centers of the two celestial bodies." "I also have a vague feeling that it should be," Michele said. "So I can also say: X equals nine tenths of D, v equals zero, and my formula becomes..." Barbicane quickly wrote down his equation on paper: v0 = 2gr [1-10r/9d-1/81(10r/dr/(dr)] Nicholl glanced at it greedily. "That's it: that's it!" he exclaimed. "Is that clear?" said Barbicane. "As clear as if written in flames!" replied Nicholl. "You two are very nice!" Michel murmured. "You understand now, don't you?" Barbicane asked him. "Do I understand?" cried Michel Ardan, "that is to say, my head has exploded!" "Therefore," continued Barbicane, "the square of v is equal to two gr times one, minus ten r of nine d, minus one eighty-one, multiplied by ten six of r, minus the ratio of d and r. r of the difference." "Now," said Nicholl, "it is only necessary to perform calculations to find the velocity of the projectile after it passes through the atmosphere." 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 with his eyes, while Michel Ardan cupped his head, which was beginning to ache, in both hands. "Well?" asked Barbicane, after a few minutes' silence. "Very well! After the calculations," Nicholl replied, "the velocity of the projectile when it leaves the atmosphere and travels to the place where the two forces of gravity are equal should be..." "It should be..." said Barbicane. "Eleven thousand and fifty-one meters." "Ah!" said Barbicane, jumping up. "What did you say?" "Eleven thousand and thirty-one meters." "Damn it!" cried the club president, making a desperate gesture. "What's the matter with you?" Michel Ardan asked in amazement. "Ask me what's wrong! The current speed has been reduced by one-third due to the friction of the air, so the initial speed should be..." "Sixteen thousand five hundred and seventy-six meters!" Nicholl replied. "Cambridge Observatory states that a muzzle velocity of 11,000 meters is sufficient. This is what propels our shells away from the Earth!" "How is it?" Nicholl asked. "How about it! This speed is not enough!" "what?" "We can't reach the zero-gravity line!" "damned!" "We couldn't even cover half the distance!" "Damn!" cried Michel Ardan, jumping up suddenly, as if the projectile were about to hit the earth. "We're going to re-land to Earth!"