题目内容
要把长为
的铁钉钉入木板中,每打击一次给予的能量为
,已知钉子在木板中遇到的阻力与钉子进入木板的深度成正比,比例系数为k。问此钉子全部进入木板需要打击几次?
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114440749185.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114440764220.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114440780507.gif)
在把钉子打入木板的过程中,钉子把得到的能量用来克服阻力做功,而阻力与钉子进入木板的深度成正比,先求出阻力的平均值,便可求得阻力做的功。
钉子在整个过程中受到的平均阻力为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114440796583.gif)
钉子克服阻力做的功为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114440811595.gif)
设全过程共打击n次,则给予钉子的总能量:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114440827594.gif)
所以![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114440780507.gif)
钉子在整个过程中受到的平均阻力为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114440796583.gif)
钉子克服阻力做的功为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114440811595.gif)
设全过程共打击n次,则给予钉子的总能量:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114440827594.gif)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114440780507.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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