题目内容
(17分)如图所示,光滑固定轨道的两端都是半径为R的四分之一圆弧,在轨道水平面上有两个质量均为m的小球B、C,B、C用一长度锁定不变的轻小弹簧栓接,弹性势能
.一质量也为m的小球A从左侧的最高点自由滑下,A滑到水平面与B碰后立即粘在一起结合成D就不再分离(碰撞时间极短).当D、C一起刚要滑到右侧最低点时,弹簧锁定解除且立即将C弹出并与弹簧分离.求
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250023310512888.png)
(1)弹簧锁定解除前瞬间,D、C速度大小
(2)弹簧锁定解除后,C第一次滑上轨道右侧圆弧部分的轨迹所对的圆心角
(3)弹簧锁定解除后,若C、D(含弹簧)每次碰撞均在水平面;求第N次碰撞结束时,C、D的速度
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002331036833.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250023310512888.png)
(1)弹簧锁定解除前瞬间,D、C速度大小
(2)弹簧锁定解除后,C第一次滑上轨道右侧圆弧部分的轨迹所对的圆心角
(3)弹簧锁定解除后,若C、D(含弹簧)每次碰撞均在水平面;求第N次碰撞结束时,C、D的速度
(1)
(2)
(3)当N为奇数时:
;当N为偶数时:![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250023311141072.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002331067696.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002331082457.png)
(3)当N为奇数时:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250023310981575.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250023311141072.png)
试题分析: (1)小球A下滑由动能定理:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002331129801.png)
A与B碰撞满足动量守恒:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002331145546.png)
由①②解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002331067696.png)
(2)由动量守恒:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002331160705.png)
机械能守恒:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250023311761206.png)
由③④⑤解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250023311921504.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002331207912.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250023312231104.png)
由⑥⑦解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002331082457.png)
(3)以左为正方向
第一次:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002331254757.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250023312701092.png)
由⑧⑨得
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250023312851170.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250023313011347.png)
第二次:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002331316928.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250023313321087.png)
由⑩??解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250023313481546.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002331363993.png)
综上,当N为奇数时:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250023310981575.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250023311141072.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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