[原创]岸上的猫和水里的老鼠

noski

<P>想到一个有趣的题：</P>
<P>有一个圆形的湖，直径100米。</P>
<P>一只老鼠在湖心处游泳，岸边来了一只猫。</P>
<P>猫很饿啊，到嘴边的美味自然不会放过，可是猫不会游泳，只好绕着湖岸追老鼠跑。</P>
<P>猫的速度是10m/s，那么，老鼠游泳速度至少要多少才能安全逃上岸呢？</P>
<P>（老鼠到岸边时，若猫没有到达老鼠的位置，老鼠便安全了）</P>
<P>&nbsp;</P>
<P>我想，老鼠大可不必沿着半径直线游回岸边去，所以，它可以游得比3.18m/s更慢一些。。。那是怎样的路线呢?</P>
<P>&nbsp;</P>
<P>&nbsp;</P>
<P>--------------------------------------------------</P>
<P>最终答案见：11# noski&nbsp;&nbsp; 29# 金眼睛&nbsp; </P>
<P>以及26# 钟七珍 所贴出的百度贴吧链接</P>
<P>--------------------------------------------------</P>
<P>&nbsp;</P>

[ 本帖最后由 noski 于 2008-10-15 19:02 编辑 ]

刚吃完

此题难度非同一般，
答案：将是一本书。
没有娱乐性，

kexin_xiao

来学习一下

psuynjw

个人认为:猫不是傻的.不管老鼠怎么跑,只要距离和猫不超过半径10米,猫都可以不动.
猫并不是非要朝着老鼠游的方向的那个方向跑,只有老鼠和猫之间的距离超过10米了,猫才需要路.本人不才,没学好数学和语文,不知道有没有描述清楚.
比如:老鼠离岸只有一米了,猫已经在离老鼠一米的岸上等了,老鼠转180度,猫没必要立刻向老鼠转的那个方向的岸上跑.

金眼睛

<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"><FONT color=#000000>也来谈谈对这道题的一些粗浅的想法，呵呵，其实大家都说得挺全面了。</FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; mso-list: l0 level1 lfo1; tab-stops: list 18.0pt"><SPAN lang=EN-US style="mso-fareast-font-family: 'Times New Roman'"><SPAN style="mso-list: Ignore"><FONT face="Times New Roman"><FONT color=#000000>1.<SPAN style="FONT: 7pt 'Times New Roman'">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </SPAN></FONT></FONT></SPAN></SPAN><FONT color=#000000><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">猫如何判断往哪个方向追：我觉得老鼠的运动方向不是猫考虑的主要问题，它要考虑老鼠在自己和圆心形成的直线的哪一侧，比如老鼠在图中的</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">D</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">点，无论运动方向如何，猫都向右侧运动。猫的主要目的是将老鼠置于线</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">OA</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">上，这样老鼠就不得不游回</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">O</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">点，从而不能逃脱。</SPAN></FONT></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; mso-list: l0 level1 lfo1; tab-stops: list 18.0pt"><SPAN lang=EN-US style="mso-fareast-font-family: 'Times New Roman'"><SPAN style="mso-list: Ignore"><FONT face="Times New Roman"><FONT color=#000000>2.<SPAN style="FONT: 7pt 'Times New Roman'">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </SPAN></FONT></FONT></SPAN></SPAN><FONT color=#000000><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">老鼠的第一段路径：老鼠的目的与猫相反，老鼠应努力将自己置于线</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">OC</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">上，假设老鼠的最大游动速度为</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">V</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">，将该速度作为保持在</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">OC</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">线上的线速度，可以算出老鼠的最大能力点，</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">OB=5V</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">，则开始的时候老鼠从</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">O</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">游到</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">B</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">可以非常悠闲，路线也是任意的，甚至可以往回游，它只需要提供一个小于</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">V</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">的线速度，从而将自己保持在线</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">OC</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">上，注意这时候</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">A</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">点是动的，线</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">OC</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">是相对于点</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">A</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">的。</SPAN></FONT></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; mso-list: l0 level1 lfo1; tab-stops: list 18.0pt"><SPAN lang=EN-US style="mso-fareast-font-family: 'Times New Roman'"><SPAN style="mso-list: Ignore"><FONT face="Times New Roman"><FONT color=#000000>3.<SPAN style="FONT: 7pt 'Times New Roman'">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </SPAN></FONT></FONT></SPAN></SPAN><FONT color=#000000><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">老鼠的第二段路径：等到了老鼠的最大能力点</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">B</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">的时候，老鼠就不能保持在</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">OC</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">上了，它应该在</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">OA</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">覆盖到自己身上的同时到达岸边。这时候老鼠先不动，观察猫，猫如果不动，老鼠就沿</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">BC</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">动，速度可以任意小，所以猫必须先动，老鼠向其相反方向启动，启动的同时，猫不会变向，因为老鼠这个瞬间在</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">OA</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">直线的左侧。</SPAN></FONT></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; mso-list: l0 level1 lfo1; tab-stops: list 18.0pt"><SPAN lang=EN-US style="mso-fareast-font-family: 'Times New Roman'"><SPAN style="mso-list: Ignore"><FONT face="Times New Roman"><FONT color=#000000>4.<SPAN style="FONT: 7pt 'Times New Roman'">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </SPAN></FONT></FONT></SPAN></SPAN><FONT color=#000000><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">关于第二段路径的方向：老鼠可以选择右侧弧</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">AC</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">中的任意一点上岸，无论是哪一点，为了保持速度最小，路径应该是直线，我计算了一下，如果猫一直不变向，老鼠沿</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">BA</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">游速度是最小的，但是，</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">F</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">点是个拐点，过了</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">F</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">点老鼠总会在路径的某点处引起猫的转向，所以楼主算得对，</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">BF</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">就是最佳路径，老鼠的速度是Vmin=2.172336282m/s。</SPAN></FONT></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; mso-list: l0 level1 lfo1; tab-stops: list 18.0pt"><FONT color=#000000><SPAN lang=EN-US style="mso-fareast-font-family: 'Times New Roman'"><SPAN style="mso-list: Ignore"><FONT face="Times New Roman">5.<SPAN style="FONT: 7pt 'Times New Roman'">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </SPAN></FONT></SPAN></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">如果老鼠选的是</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">C</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">点，结果就是</SPAN><SPAN lang=EN-US><FONT face="Times New Roman">10/(pi+1)</FONT></SPAN><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">，呵呵！或者老鼠如果水性好，可以跟猫比比耐力啊，嘿嘿！</SPAN></FONT></P>

[ 本帖最后由 金眼睛 于 2008-5-25 22:00 编辑 ]

noski

在百度贴吧的讨论结果，是说11楼我的那个答案是最优结果喽？

bbshanwei

题目都牵扯到刘翔了！！不过那个螺旋上的点有点难理解。

钟七珍

　
　　我认为，11楼的答案就是最佳方案。但是证明较难。
　　有关此问题的讨论，可参见：http://tieba.baidu.com/f?z=34111 ... 6%A8%CA%C6&pn=0