（1）船的实际运动是：水流的运动和船相对静水的运动的合运动。
（2）三种速度：v1（船在静水中的速度）、v2（水流速）、v（船的实际速度）。
（3）三种情景：
①过河时间最短：船头正对河岸时，渡河时间最短，![](https://wkrtcs.bdimg.com/rtcs/image?w=52&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"52","h":"40","dataType":"png","c":"word/media/image1_1.png","imgOriW":1280,"imgOriH":1024})
（d为河宽）。
②过河路径最短（![](https://wkrtcs.bdimg.com/rtcs/image?w=46&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"46","h":"20","dataType":"png","c":"word/media/image2_1.png","imgOriW":1120,"imgOriH":512})
时）：合速度垂直于河岸，航程最短，。
③过河路径最短（![](https://wkrtcs.bdimg.com/rtcs/image?w=46&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"46","h":"20","dataType":"png","c":"word/media/image4_1.png","imgOriW":1152,"imgOriH":512})
时）：合速度不可能垂直于河岸，无法垂直渡河。确定方法如下：如图所示，以v2矢量末端为圆心，以v1矢量的大小为半径画弧，从v2矢量的始端向圆弧作切线，则合速度沿此切线方向航程最短。
![](https://wkrtcs.bdimg.com/rtcs/image?w=222&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"222","h":"118","dataType":"png","c":"word/media/image5.png","imgOriW":224,"imgOriH":128})
由图可知：![](https://wkrtcs.bdimg.com/rtcs/image?w=63&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"63","h":"40","dataType":"png","c":"word/media/image6_1.png","imgOriW":1568,"imgOriH":1024})
，最短航程：
问题3：小船渡河的最值问题：
一小船渡河，河宽d=180m，水流速度![](https://wkrtcs.bdimg.com/rtcs/image?w=78&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"78","h":"20","dataType":"png","c":"word/media/image8_1.png","imgOriW":1888,"imgOriH":512})
。
（1）若船在静水中的速度为![](https://wkrtcs.bdimg.com/rtcs/image?w=69&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"69","h":"20","dataType":"png","c":"word/media/image9_1.png","imgOriW":1664,"imgOriH":512})
，求：
①欲使船在最短的时间内渡河，船头应朝什么方向？用多长时间？位移是多少？
②欲使船渡河的航程最短，船头应朝什么方向？用多长时间？位移是多少？
（2）若船在静水中的速度![](https://wkrtcs.bdimg.com/rtcs/image?w=78&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"78","h":"20","dataType":"png","c":"word/media/image10_1.png","imgOriW":1888,"imgOriH":512})
，要使船渡河的航程最短，船头应朝什么方向？用多长时间？位移是多少？
解析：将船实际的速度（合速度）分解为垂直河岸方向和平行河岸方向的两个分速度，垂直分速度影响渡河的时间，而平行分速度只影响平行河岸方向的位移。
（1）若![](https://wkrtcs.bdimg.com/rtcs/image?w=69&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"69","h":"20","dataType":"png","c":"word/media/image11_1.png","imgOriW":1664,"imgOriH":512})
。
①欲使船在最短时间内渡河，船头应朝垂直河岸方向。
当船头垂直河岸时，如图所示，合速度为倾斜方向，垂直分速度为![](https://wkrtcs.bdimg.com/rtcs/image?w=69&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"69","h":"20","dataType":"png","c":"word/media/image12_1.png","imgOriW":1664,"imgOriH":512})
。
![](https://wkrtcs.bdimg.com/rtcs/image?w=207&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"207","h":"181","dataType":"png","c":"word/media/image13.png","imgOriW":210,"imgOriH":184})
![](https://wkrtcs.bdimg.com/rtcs/image?w=121&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"121","h":"40","dataType":"png","c":"word/media/image14_1.png","imgOriW":2266,"imgOriH":788})
![](https://wkrtcs.bdimg.com/rtcs/image?w=167&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"167","h":"37","dataType":"png","c":"word/media/image15_1.png","imgOriW":2808,"imgOriH":646})
![](https://wkrtcs.bdimg.com/rtcs/image?w=107&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"107","h":"23","dataType":"png","c":"word/media/image16_1.png","imgOriW":2624,"imgOriH":608})
②欲使船渡河航程最短，应垂直河岸渡河，船头应朝上游与河岸成某一角度α。
垂直河岸过河这就要求![](https://wkrtcs.bdimg.com/rtcs/image?w=55&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"55","h":"23","dataType":"png","c":"word/media/image17_1.png","imgOriW":1344,"imgOriH":544})
，所以船头应向上游偏转一定角度，如图所示，有得。
![](https://wkrtcs.bdimg.com/rtcs/image?w=184&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"184","h":"150","dataType":"png","c":"word/media/image20.png","imgOriW":186,"imgOriH":151})
所以当船头向上游偏30°时航程最短。
s=d=180m.
![](https://wkrtcs.bdimg.com/rtcs/image?w=193&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"193","h":"58","dataType":"png","c":"word/media/image21_1.png","imgOriW":2497,"imgOriH":753})
（2）若![](https://wkrtcs.bdimg.com/rtcs/image?w=81&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"81","h":"20","dataType":"png","c":"word/media/image22_1.png","imgOriW":1952,"imgOriH":512})
与（1）中②不同，因为船速小于水速，所以船一定向下游漂移，设合速度方向与河岸下游方向夹角为α，则航程![](https://wkrtcs.bdimg.com/rtcs/image?w=55&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"55","h":"37","dataType":"png","c":"word/media/image23_1.png","imgOriW":1344,"imgOriH":928})
，欲使航程最短，需α最大，如图所示，由出发点A作出v1矢量，以v1矢量末端为圆心，v2大小为半径作圆，A点与圆周上某点的连线即为合速度方向，欲使v合与水平方向夹角最大，应使v合与圆相切，即。
![](https://wkrtcs.bdimg.com/rtcs/image?w=196&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"196","h":"181","dataType":"png","c":"word/media/image25.png","imgOriW":197,"imgOriH":182})
![](https://wkrtcs.bdimg.com/rtcs/image?w=124&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"124","h":"40","dataType":"png","c":"word/media/image26_1.png","imgOriW":2293,"imgOriH":781})
，得α＝37°。
所以船头应向上游偏37°。
![](https://wkrtcs.bdimg.com/rtcs/image?w=170&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"170","h":"40","dataType":"png","c":"word/media/image27_1.png","imgOriW":2716,"imgOriH":674})
。
![](https://wkrtcs.bdimg.com/rtcs/image?w=265&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"265","h":"23","dataType":"png","c":"word/media/image28_1.png","imgOriW":4597,"imgOriH":393})
。
变式4：
一艘小艇从河岸A处出发渡河，小艇保持与河岸垂直方向行驶，经过10min到达正对岸下游120m的C处，如图所示，如果小艇保持原来的速度逆水斜向上游与河岸成α角方向行驶，则经过12.5min恰好到达正对岸的B处，求：河的宽度。
![](https://wkrtcs.bdimg.com/rtcs/image?w=181&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"181","h":"107","dataType":"png","c":"word/media/image29.png","imgOriW":183,"imgOriH":108})
解析：设河宽为d，河水流速为![](https://wkrtcs.bdimg.com/rtcs/image?w=23&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"23","h":"23","dataType":"png","c":"word/media/image30_1.png","imgOriW":544,"imgOriH":544})
，船速为，船两次运动速度合成如图所示。
![](https://wkrtcs.bdimg.com/rtcs/image?w=423&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"423","h":"156","dataType":"png","c":"word/media/image32.png","imgOriW":426,"imgOriH":156})
依题意有：![](https://wkrtcs.bdimg.com/rtcs/image?w=135&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"135","h":"23","dataType":"png","c":"word/media/image33_1.png","imgOriW":3296,"imgOriH":544})
①
![](https://wkrtcs.bdimg.com/rtcs/image?w=75&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"75","h":"23","dataType":"png","c":"word/media/image34_1.png","imgOriW":1856,"imgOriH":608})
②
![](https://wkrtcs.bdimg.com/rtcs/image?w=72&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"72","h":"43","dataType":"png","c":"word/media/image35_1.png","imgOriW":1760,"imgOriH":1088})
由②可得![](https://wkrtcs.bdimg.com/rtcs/image?w=95&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"95","h":"23","dataType":"png","c":"word/media/image36_1.png","imgOriW":2336,"imgOriH":544})
由①得![](https://wkrtcs.bdimg.com/rtcs/image?w=66&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"66","h":"14","dataType":"png","c":"word/media/image37_1.png","imgOriW":1600,"imgOriH":416})
，故，
河宽![](https://wkrtcs.bdimg.com/rtcs/image?w=184&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"184","h":"37","dataType":"png","c":"word/media/image39_1.png","imgOriW":2971,"imgOriH":615})
。
变式5：
船在静水中的航速为v1，水流的速度为v2。为使船行驶到河正对岸的码头，则v1相对v2的方向应为（ ）
![](https://wkrtcs.bdimg.com/rtcs/image?w=556&md5sum=be413c4b1f9384713a4771e396e61df6&sign=0c1246c707&rtcs_flag=1&rtcs_ver=3.3&l=webapp&bucketNum=47&ipr={"t":"img","w":"556","h":"95","dataType":"png","c":"word/media/image40.png","imgOriW":557,"imgOriH":96})
答案：C
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