题目内容
荷兰科学家惠更斯在研究物体碰撞问题时做出了突出的贡献.惠更斯所做的碰撞实验可简化为:三个质量分别为m
、m
、m
的小球,半径相同,并排悬挂在长度均为L的三根平行绳子上,彼此相互接触。现把质量为m
的小球拉开,上升到H高处释放,如图所示,已知各球间碰撞时同时满足动量守恒定律和机械能守恒定律,且碰撞时间极短,H远小于L,不计空气阻力。
小题1:若三个球的质量相同,则发生碰撞的两球速度交换,试求此时系统的运动周期。
小题2:若三个球的质量不同,要使球1与球2、球2与球3相碰之后,三个球具有同样的动量,则m
∶m
∶m
应为多少?它们上升的高度分别为多少?![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241148316763077.jpg)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831364127.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831411129.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831426180.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831442127.gif)
小题1:若三个球的质量相同,则发生碰撞的两球速度交换,试求此时系统的运动周期。
小题2:若三个球的质量不同,要使球1与球2、球2与球3相碰之后,三个球具有同样的动量,则m
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831364127.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831411129.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831426180.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241148316763077.jpg)
小题1:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831692553.gif)
小题2:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831879219.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831894325.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831910326.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831926347.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831941373.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831957311.gif)
小题1:球1与球2、球2与球3碰撞后速度互换,球3以球1碰球2前瞬间的速度开始上升到H高处,然后再摆回来与球2、球2与球1碰撞,使球1上升到H高处,此后,系统做到周期性运动,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831957896.gif)
由此可知系统的运动周期为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831692553.gif)
小题2:由题意知三球碰后的动量均相同,设为p,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114832050476.gif)
根据机械能守恒定律,对于球2与球3碰撞的情况应有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114832066841.gif)
由此得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831894325.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831910326.gif)
球1与球2碰前的动量为3p,根据机械能守恒定律有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114832128859.gif)
由此得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831879219.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831894325.gif)
从而可得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831879219.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831894325.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831910326.gif)
设三球碰后上升的高度分别为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114832253355.gif)
球1碰前动能
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114832269244.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114832284401.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114832269244.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114832456497.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114832472226.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114832487507.gif)
球1碰后动能
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114832269244.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114832518417.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114832269244.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114832550406.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114832550564.gif)
从而可得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831926347.gif)
同理可得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831941373.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824114831957311.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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