小船渡河问题的拓展

EXPANSION OF THE SMALL BOAT CROSSING THE RIVER PROBLEM

在考虑河水流速与距河岸距离的平方成正比条件下,首先详细地分析了当小船渡河角度一定时的渡河过程,给出了若干运动学方程,然后通过求极值的方式解决了最短过河时间与最小过河位移问题。同时还考虑小船渡河时船头始终指向定点的运动情况,求得了其轨迹方程与过河时间。并且得出了无论船速大小都最终会行驶到定点的结论。此外还结合Maple软件绘制出以上两种情况下的小船渡河动画,从而以定性分析与定量计算相结合的方式对该问题进行较为完整的诠释。

Under the condition that the river velocity is proportional to the square of the distance from the river bank, the process of crossing the river when the boat is crossing the river at a certain angle is analyzed in detail, and several kinematic equations are given. Then the shortest crossing time and the smallest crossing displacement are solved by finding the extremum. At the same time, the motion of the boat’s bow always pointing to the fixed point is considered, and the trajectory equation and crossing time are obtained. It is concluded that no matter the speed of the ship, it will eventually travel to a fixed point. In addition, Maple software is used to draw the animation of boat crossing in the above two situations, so as to explain the problem in a more complete way by combining qualitative analysis and quantitative calculation.