本文基于第五版《数学模型》第二章第九节“扬帆远航”模型,探究帆船在航行过程中如何调整航向和帆的朝向以实现最快速度到达终点。首先分析了帆船在东北和东南两个方向上受到风力的作用,并建立相应的数学模型描述航向、帆的朝向与帆船...本文基于第五版《数学模型》第二章第九节“扬帆远航”模型,探究帆船在航行过程中如何调整航向和帆的朝向以实现最快速度到达终点。首先分析了帆船在东北和东南两个方向上受到风力的作用,并建立相应的数学模型描述航向、帆的朝向与帆船速度之间的关系。其次,求解了东北和东南方向最优的航向和帆的朝向,同时通过数值计算建立了航向和帆的朝向与帆船转东南方向夹角之间的关系。最后,确定了该夹角的取值范围及使帆船到达终点用时最少的最优夹角。This article is based on the “Sailing Far Away” model in Section 9 of Chapter 2 of the fifth edition of “Mathematical Models”, exploring how sailboat adjusts its heading and sail orientation during navigation to achieve the fastest speed to reach the destination. Firstly, it analyzes the effect of wind on sailboat in the northeast and southeast directions and establishes corresponding mathematical models to describe the relationship between heading, sail orientation and boat speed. Secondly, the optimal heading and sail orientation in the northeast and southeast directions were solved, and establishes the relationship between heading, sail orientation, and the angle at which the boat turns towards the southeast direction through numerical calculations. Finally, it defines the range of the angle and identifies the optimal angle that minimizes the time for the sailboat to reach the destination.展开更多
文摘本文基于第五版《数学模型》第二章第九节“扬帆远航”模型,探究帆船在航行过程中如何调整航向和帆的朝向以实现最快速度到达终点。首先分析了帆船在东北和东南两个方向上受到风力的作用,并建立相应的数学模型描述航向、帆的朝向与帆船速度之间的关系。其次,求解了东北和东南方向最优的航向和帆的朝向,同时通过数值计算建立了航向和帆的朝向与帆船转东南方向夹角之间的关系。最后,确定了该夹角的取值范围及使帆船到达终点用时最少的最优夹角。This article is based on the “Sailing Far Away” model in Section 9 of Chapter 2 of the fifth edition of “Mathematical Models”, exploring how sailboat adjusts its heading and sail orientation during navigation to achieve the fastest speed to reach the destination. Firstly, it analyzes the effect of wind on sailboat in the northeast and southeast directions and establishes corresponding mathematical models to describe the relationship between heading, sail orientation and boat speed. Secondly, the optimal heading and sail orientation in the northeast and southeast directions were solved, and establishes the relationship between heading, sail orientation, and the angle at which the boat turns towards the southeast direction through numerical calculations. Finally, it defines the range of the angle and identifies the optimal angle that minimizes the time for the sailboat to reach the destination.
