[原创]岸上的猫和水里的老鼠

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 ]