中文

2022年亚洲及太平洋地区建模数学竞赛

问题E

有多少颗核弹可以摧毁地球？

1945年8月6日，第二次世界大战已接近尾声。为了尽快结束战争，美国在日本广岛投下了名为 "小男孩" 的第二颗原子弹。这样一颗原子弹炸死了广岛的200000人，广岛的所有建筑都倒塌了。这是人类历史上第一次将原子弹用于实践，这也使许多人第一次认识到原子弹的可怕威力。

图1. 原子弹爆炸的模拟图。

核武器是指与核反应有关的巨大杀伤性武器，包括氢弹、原子弹、中子弹等。核武器是人类有史以来开发的最强大的武器之一，它常常让人想起毁天灭地的场景。原子弹的瞬时爆炸温度可达数千万度。原子弹的爆炸当量约为几万至几十万吨TNT当量。一颗原子弹的爆炸和它的辐射区域可以摧毁一个城市。

由于原子弹的可怕威力，许多国家希望用它来威慑其他国家，保护自己的国家免受外国入侵。第二次世界大战之后，世界各国开始疯狂地研究和制造原子弹，甚至创造了诸如 "大伊万" 这样的破坏性武器。

"大伊万" 是目前世界上已知威力最大的核弹，也就是苏联时期制造的 "沙皇炸弹"。它不仅是世界上最强大的核弹，也是最大的核弹。"沙皇炸弹" 到底有多大？数据显示，其长度为8米，直径为2.1米，重量高达27吨，设计TNT当量为50兆吨。苏联最初计划设计的当量为100兆吨，但由于破坏力太大，无法找到合适的试验场地，所以威力减少了一半。即便如此，沙皇炸弹仍然是世界上最强大的核武器。 "小男孩" 的爆炸威力约为14000吨TNT当量，使广岛成为一片废墟，而 "沙皇炸弹" 的威力可与几十个甚至几百个 "小男孩" 相比。在苏联新地岛试爆 "沙皇炸弹" 后，甚至在远在美国的一些州，都发现了约5级的地震，导致欧亚大陆向南移动了9毫米。

许多人认为，这些核武器可以多次摧毁地球。情况真的是这样吗？就人类研制的核武器的威力而言，如果摧毁地球，并不是指可以把地球炸成碎片，而是指人类和地球上的生物的生存环境被破坏了。

APMCM组委会要求你的团队解决目前的报告和未来的核武器预测。他们提供了2022_APMCM_E_Data.csv的数据集，以帮助你的研究。请收集相应的数据，建立一个数学模型并回答以下问题。

要求

1. 基本数据分析

a) 哪些国家曾经拥有过核武器？

b) 在过去20年中，哪个国家的核武器库存减少或增加最多？

c) 在哪五年中，核武器试验发生得最多？

d) 在过去10年中，哪个国家的核武器研究最为活跃？

e) 哪个国家从 "不考虑核武器" 到 "拥有核武器" 的转变最快？

2. 预测核武器的数量

a) 根据附件中的数据或你收集的数据，建立一个数学模型来预测核武器的数量，并预测未来100年拥有核武器的国家。

b) 预测未来100年核武器数量的变化趋势，2123年的核武器总数，以及每个国家的核武器数量。

3. 保护我们的地球

a) 建立核武器引爆位置的数学模型，并计算出至少需要多少枚核弹才能摧毁地球？

b) 根据该数学模型，目前拥有的核弹的最大破坏力是多少？它是否足以摧毁地球？

c) 为了保护地球和我们赖以生存的环境，世界上的核弹总数应该限制在多少？而已经拥有核武器的国家在理论上应该限制在多少？

4. 准备一篇非技术性的文章（最多1页）

请给联合国（U.N.）写一篇非技术性的文章（最多1页），解释你们团队的发现，并为所有国家提出几条建议。

附文

数据来源。数据中的我们的世界： https://ourworldindata.org/nuclear-weapons

2022_APMCM_E_Data.xlsx

立场表状态。0-不考虑, 1-考虑, 2-追求, 3-拥有

思路见新一篇文章

英文

2022 Asia and Pacific Mathematical Contest in Modeling

Problem E

How Many Nuclear Bombs can Destroy the Earth?

On August 6, 1945, the Second World War was drawing to a close. In order to end the war as soon as possible, the United States dropped the next atomic bomb called "Little Boy" in Hiroshima, Japan. Such an atomic bomb killed 200000 people in Hiroshima, and all buildings in Hiroshima collapsed. This is the first time in human history that the atomic bomb has been put into practice, and it also makes many people realize the terrible power of the atomic bomb for the first time.

Figure 1. Simulation diagram of atomic bomb explosion.

Nuclear weapons refer to huge lethal weapons related to nuclear reaction, including hydrogen bombs, atomic bombs, neutron bombs, etc. Nuclear weapons are one of the most powerful weapons ever developed by human beings, and they often remind people of the scene of destroying heaven and earth. The instantaneous explosion temperature of an atomic bomb can reach tens of millions of degrees. The explosive yield of the atomic bomb is about tens of thousands to hundreds of thousands of tons of TNT equivalent. The explosion of an atomic bomb and the area of its radiation can destroy a city.

Because of the terrible power of the atomic bomb, many countries hope to use it to deter other countries and protect their countries from foreign invasion. After World War II, all countries in the world began to research and manufacture atomic bombs crazily, and even created destructive weapons such as "Big Ivan".

The "Big Ivan" is the most powerful nuclear bomb known in the world at present, that is, the "czar bomb" built in the Soviet period. It is not only the most powerful nuclear bomb in the world, but also the largest nuclear bomb. How big is the "Tsar Bomb"? The data shows that its length is 8 meters, diameter is 2.1 meters, weight is up to 27 tons, and design TNT equivalent is 50 megatons. The Soviet Union originally planned to design an equivalent of 100 megatons, but because the destructive force was too great to find a suitable test site, the power was reduced by half. Even so, the Czar Bomb is still the most powerful nuclear weapon in the world. The explosive power of the "little boy" is about 14000 tons of TNT equivalent, making Hiroshima a ruin, while the power of the "tsar bomb" is comparable to dozens or even hundreds of "little boys". After the test explosion of the "Tsar Bomb" on the Soviet Union's Xindi Island, even in some states as far away as the United States, an earthquake of about magnitude 5 was detected, resulting in a 9 mm southward movement of Eurasia.

Many people believe that these nuclear weapons can destroy the earth several times. Is this really the case? As far as the power of nuclear weapons developed by mankind is concerned, if they destroy the earth, they do not mean that they can blow the earth into pieces, but that the living environment of human beings and creatures on the earth has been destroyed.

The APMCM Organizing Committee requires your team to address the current report and future nuclear weapons projections. They provided data set 2022_APMCM_E_Data.csv to help you with your research. Please collect the corresponding data, establish a mathematical model and answer the following questions.

Requirements

1. Basic data analysis

a) Which countries have ever possessed nuclear weapons?

b) Which country has the largest reduction or increase in its nuclear weapons stockpiles in the last 20 years?

c) During which five years did nuclear weapon tests occur the most?

d) Which country has been the most active in nuclear weapons research in the last 10 years?

e) Which country has made the fastest transition from "not considering nuclear weapons" to "possessing nuclear weapons"?

2. Predict the number of nuclear weapons

a) According to the attached data or the data you collected, establish a mathematical model to predict the number of nuclear weapons, and predict the countries with nuclear weapons in the next 100 years;

b) Predict the change trend of the number of nuclear weapons in the next 100 years, the total number of nuclear weapons in 2123, and the number of nuclear weapons in each country.

3. Protect our planet

a) Establish an mathematical model for the detonation position of nuclear weapons, and calculate how many nuclear bombs are required at least to destroy the earth?

b) According to the mathematical model, what is the maximum destructive power of the nuclear bomb currently possessed? Is it enough to destroy the earth?

c) In order to protect the earth and the environment on which we live, what should the total number of nuclear bombs in the world be limited to, and what should the countries that already have nuclear weapons be limited to theoretically?

4. Prepare a non-technical article (1 page maximum)

Please write a non-technical article (1 page at most) to the United Nations (U.N.), explaining your team's findings and putting forward several suggestions for all countries.

Attachment

Data Source: Our World in Data https://ourworldindata.org/nuclear-weapons