题目内容
如图所示,物块
、
、
的质量分别为
、
、
,并均可视为质点,它们间有
关系。三物块用轻绳通过滑轮连接,物块
与
间的距离和
到地面的距离均是
。若
与地面、
与
相碰后速度立即减为零,
与
相碰后粘合在一起。(设
距离滑轮足够远且不计一切阻力)。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241211175684609.jpg)
(1)求物块
刚着地时的速度大小?
(2)若使物块
不与
相碰,则
应满足什么条件?
(3)若
时,求物块
由最初位置上升的最大高度?
(4)若在(3)中物块
由最高位置下落,拉紧轻绳后继续下落,求物块
拉紧轻绳后下落的最远距离?
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117271200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117302206.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117318205.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117334327.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117349332.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117365204.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117380466.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117302206.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117318205.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117318205.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117443197.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117318205.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117302206.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117318205.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117302206.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117318205.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117271200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241211175684609.jpg)
(1)求物块
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117318205.gif)
(2)若使物块
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117302206.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117318205.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117630401.gif)
(3)若
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117646406.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117271200.gif)
(4)若在(3)中物块
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117271200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117271200.gif)
①
②
③
④![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117770341.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117724860.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117739517.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117755625.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117770341.gif)
解:①设
到达地面时三者速度大小为V1,![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117802878.gif)
解得![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117724860.gif)
②设此后
到达地面时速度恰好为零。有:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117848904.gif)
解得:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117864477.gif)
因此应满足:
时,物块
不能着地。
③若
时,设
到达地面时三者速度大小为V2,
,再设
运动到
到达地面时速度大小为
,有:
,
此后
物块还能上升的高度为
,![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121118457619.gif)
可得
物块上升的最大高度为![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117755625.gif)
④物块
下落距离
时,拉紧细线,设此时物块
速度大小为
,有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121118582698.gif)
此时由动量守恒定律得
、
三者有大小相等的速度设为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121118644617.gif)
设
拉紧细线后下落的最远距离为
:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121118706913.gif)
由以上几式可得:![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117770341.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117318205.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117802878.gif)
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117724860.gif)
②设此后
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117302206.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117848904.gif)
解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117864477.gif)
因此应满足:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117739517.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117302206.gif)
③若
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117646406.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117318205.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121118254869.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121118270235.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117302206.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121118301202.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408241211183161180.gif)
此后
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117271200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121118348200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121118457619.gif)
可得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117271200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117755625.gif)
④物块
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117271200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121118519462.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117271200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121118550202.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121118582698.gif)
此时由动量守恒定律得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117271200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121118597241.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121118613201.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121118644617.gif)
设
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117271200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121118675182.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121118706913.gif)
由以上几式可得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824121117770341.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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