题目内容
(10分)设雨点下落过程受到的空气阻力与雨点的横截面积S成正比,与雨点下落的速度v的平方成正比,即
(其中k为比例系数).雨点接近地面时近似看做匀速直线运动,重力加速度为g.若把雨点看做球形,其半径为r,球的体积为
,设雨点的密度为
,求:
(1)每个雨点最终的运动速度
(用
、r、g、k表示);
(2)雨点的速度达到
时,雨点的加速度a为多大?
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001703749585.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001703905586.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001703921311.png)
(1)每个雨点最终的运动速度
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001703937367.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001703921311.png)
(2)雨点的速度达到
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001703968499.png)
(1)
(2)![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001703999477.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001703983913.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001703999477.png)
试题分析:(1)由于雨点接近地面时近似匀速直线运动
因此
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001704015511.png)
而
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001704030867.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001703749585.png)
而
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001704061531.png)
联立得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001703983913.png)
(2)根据牛顿第二定律
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001704077634.png)
而此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250017043111135.png)
整理得
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825001704327563.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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