Quan787：创建页面，内容为“{{由ai生成}} ==英文题目== '''T3. Asteroid (10 points)''' A peculiar asteroid of mass, $m$, was spotted at a distance, $d$, from a star with mass, $M$.…”

2025-03-06T14:54:57Z

创建页面，内容为“{{由ai生成}} ==英文题目== '''T3. Asteroid (10 points)''' A peculiar asteroid of mass, $m$, was spotted at a distance, $d$, from a star with mass, $M$.…”

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{{由ai生成}}
==英文题目==

'''T3. Asteroid (10 points)'''

A peculiar asteroid of mass, $m$, was spotted at a distance, $d$, from a star with mass, $M$. The magnitude of the asteroid’s velocity at the time of the observation was $v = \sqrt{\frac{GM}{d}}$, where $G$ is the universal gravitational constant. The distance $d$ is much larger than the radius of the star.

For both of the following items, express your answers in terms of $M$, $d$, and physical or mathematical constants.

(a) (8 points) If the asteroid is initially moving exactly towards the star, how long will it take for it to collide with the star?

(b) (2 points) If the asteroid is instead initially moving exactly away from the star, how long will it now take for it to collide with the star?

==中文翻译==

'''T3. 小行星（10分）'''

一颗质量为$m$的特殊小行星在距离质量为$M$的恒星$d$处被发现。观测时小行星的速度大小为$v = \sqrt{\frac{GM}{d}}$，其中$G$为万有引力常数。距离$d$远大于恒星半径。

对以下两问，答案需用$M$、$d$和物理/数学常数表示。

(a) (8分) 若小行星初始直接朝向恒星运动，它需要多长时间会与恒星碰撞？

(b) (2分) 若小行星初始直接背离恒星运动，它需要多长时间会与恒星碰撞？

== 官方解答 ==

(a) 该情况下小行星将沿退化椭圆轨道运动。根据开普勒第三定律：

$$T = 2\pi d \sqrt{\frac{d}{GM}}$$

利用开普勒第二定律计算碰撞时间：

$$\Delta t = \left( \frac{\pi}{2} - 1 \right) d \sqrt{\frac{d}{GM}}$$

(b) 当小行星初始背离恒星运动时，需遍历轨道面积II和III后返回：

$$\Delta t = \frac{\pi}{2} d \sqrt{\frac{d}{GM}}$$

[[分类:天体力学]]
[[分类:由ai生成]]

2024年IOAA理论第3题-小行星 - 版本历史

Quan787：创建页面，内容为“{{由ai生成}} ==英文题目== '''T3. Asteroid (10 points)''' A peculiar asteroid of mass, \(m\), was spotted at a distance, \(d\), from a star with mass, \(M\).…”