美国夢沃学校(上海)2016年春分话测纬活动

A physics assignment to measure latitude

电影《金蝉脱壳》中硬汉史泰龙深陷囹圄，想要求救却不知身在何处。神奇的是，硬汉竟然自制了仿佛是山寨六分仪的东西，测量了所在地的纬度，最终招来救兵，脱得牢笼，坏人也终有恶报。不久前，美国梦沃学校(上海)物理Honor班的同学们也领到了史泰龙类似的任务：在正午通过测量太阳的头顶夹角来估算当地纬度。

In the movie “Escape,” tough guy Sylvester Stallone was held in prison. He wanted to ask for help but he didn’t know where he was. Amazingly, he makes something like primitive Sextant to measure the latitude, and was ultimately able to summon reinforcements. He got rescued, and the perpetrators were punished. Not long ago the students from Honors Physics were assigned a similar task as Stallone: estimate the local latitude by measuring the angle between the Sun and a vertically suspended plumb.

每天正午之时，太阳当空，假如你在地上垂直竖一把尺，这时它的影子肯定是一天中最短的。测量此时影子的长度，计算影长和尺长的比率就得到太阳头顶夹角的正切函数，就能得到这个角度了。

When you stand a ruler vertically on the ground at noon time, its shadow should be the shortest of the day. If you measure the length of the shadow, calculate the ratio of shadow's length to the ruler's length, you will get the tangent of the angle.

可是这个角度和纬度是什么关系呢?

But the point is: what does it matter with latitude?

我们先来看看纬度是怎样定义的： 假设我们垂直竖起的这把尺就像星球大战里的光剑一样向下延长延长……，因为重力的方向总是指向地心的，所以最后正好到达地心。然后我们象切西瓜一样，沿着南北极把地球一切两半。好了，现在看清楚了：在这个地球切面上，我们深入地心的这把光剑和赤道线的夹角就是我们的纬度啊!

Let's have a look at how latitude is defined. Assuming the vertical ruler is extending like a light saber in Star Wars, as gravity always points to the center of the Earth, in the end it will reach the center. Then we cut the earth into halves like a watermelon, along the north-south axis. Now it becomes clearer: on this plane, the angle between the light from the light saber and the Equator is our latitude!

同学们又问了：我们可以在这个切面上做一个到赤道线的垂线吗?垂线和地球半径的比率不就是纬度的正弦值。可这个垂线长度怎么测呢?

The students asked: can we make a projection line to the Equator line on a plane? Measure the length of the projection line， and the ratio over the Earth radius will be the sine of the latitude. The Problem is: how to measure it?

这个比较困难，因为我们没法打一个垂直于赤道线的洞并测量它的深度，但是可以呼唤一下欧几里德!

This is more difficult because we cannot dig a hole perpendicular to the Equator and measure the depth. But we can ask Euclid!

欧几里德的平行线定理说：如果两条直线平行，被第三条直线所截，内错角相等。如果有一条线和赤道线平行，而且经过尺的顶端，这条线和尺的夹角就等于纬度。

Euclidean theorem about parallel lines states that given two parallel straight lines, when crossed by the third line, the alternate angles are equal. If there is a line passing the top of the ruler, and parallel to the equator, then the angle between the line and the ruler is equal to the latitude here.

可是上哪找这样的一条平行线呢?我们需要太阳的帮助!

However, where is the parallel line? We need the sun's help!

春夏秋冬，四季轮回，太阳直射地球的地方在北纬23度5和南纬23度5之间周而复始。

As the seasons change over the course of the year, the subsolar point will move between 23.5 degrees north and 23.5 degrees south.

2016年的3月20日，正是春分之时，太阳正好直射在地球赤道之上。在正午日影最短之时，太阳光如能穿过赤道，将直达地心!(这个时候如果你在赤道上竖根棍子，是没有影子的)因为相比地球的直径(大约一万二千公里)，太阳距离我们太远了(大约一亿五千万公里)。所以我们可以近似地认为太阳光是平行的。有了太阳光作为平行线，我们就可以测量太阳和垂直竖立直尺的夹角，而这个角等于我们所在地的纬度!

March 20, 2016, the Spring Equinox occurs: the Sun's rays hit the Earth on the equator. Now when the shadow is the shortest at noon, if the Sun's rays continued through the Equatorial line, they would reach the exact center of the earth! (If your put up a stick perpendicular to the Equator at this time, there is no shadow) Compared to the diameter of the Earth (about 12,000 km), the Sun is too far away, approximately 150 million km. So we can presume that the Sun's rays are parallel. With sun rays as parallel lines, we can measure the angle between the Sun and the vertical ruler, and this angle is equal to our latitude!

问题又来了，如果太阳没有直射在地球赤道上，我们该怎么办呢?

The problem arises, if the subsolar point is not at the Earth Equator, what should we do?

天文学家已经计算过了一年中每天太阳直射在地球上的纬度，根据太阳直射在地球的南北半球不同纬度，就可以通过头顶夹角和直射纬度计算出我们的纬度。

Astronomers have calculated the latitude of subsolar point every day of the year on Earth. According to the latitude of the subsolar point in the northern or southern hemisphere, we can then calculate our latitude.

Isabellae同学用环保材料制作了一个测量设备，由测量尺，垂线，量角器来测量夹角。

Isabellae made a measuring device with eco-friendly materials, using a measuring stick, a plumb, and a protractor to measure the angle.

同学们对测量结果进行了演算，最终估算出了自己所在地的纬度。

Students calculated the measured data, and finally estimate the latitude of their location.

物理学习，可不是简单的背公式，记定律。通过认识物体的运动规律，学习科学分析的方法，掌握解决问题的思路，这才是物理学习的精髓。测量纬度，是同学们发挥空间想象力，运用所学的天文地理，平面几何和三角函数知识来解决问题的一次演练。坚持不断地练习，就会为后续的大学学习打下良好的基础。

Learning physics is not simply to memorize equations and laws. By analyzing the motion of objects, students can learn scientific methods and problem solving strategies. This is the essence of learning. Measuring latitude creates the opportunity to truely use problem solving skills, utilize students' spatial imagination and knowledge from astronomy, geography, geometry and trigonometry. Such opportunities for our students will provide them with an excellent foundation for their future studies in college.

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