math - 你如何解决这个台球游戏？

Question

我知道答案是 B，但我不清楚为什么会这样。如果有人能善意地解释寻找答案的过程或可能展示一个模拟，那就太棒了。

谢谢你。

Answer 1

您可以沿着球的路径穿过桌子。让这件事变得简单的一点是，球从 45° 的方向开始。因此，所有的碰撞角度都是 45°。因此，您只需反转球方向矢量的一个分量。

这是一些 C# 示例代码。坐标系的原点位于表格的左下角。球的位置在其边界框的左下角测量：

int ballX = 0;
int ballY = 0;
int ballWidth = 5;
int tableWidth = 230;
int tableHeight = 130;

int directionX = 1;
int directionY = 1;

while(true)
{
    //the distances that the ball could travel until it collides with a vertical or horizontal border, respectively
    int travelDistanceX, travelDistanceY;
    if (directionX > 0)
        travelDistanceX = tableWidth - ballWidth - ballX;
    else
        travelDistanceX = ballX;

    if (directionY > 0)
        travelDistanceY = tableHeight - ballWidth - ballY;
    else
        travelDistanceY = ballY;

    if(travelDistanceX == travelDistanceY)
    {
        //we found the target pocket:
        Console.WriteLine("Target is located at {0}/{1}.", ballX + travelDistanceX * directionX, ballY + travelDistanceY * directionY);
        break;
    }

    if(travelDistanceX < travelDistanceY)
    {
        //collision with the vertical borders
        ballX += travelDistanceX * directionX;
        ballY += travelDistanceX * directionY;
        directionX *= -1;
    }
    else
    {
        //collision with the horizontal borders
        ballX += travelDistanceY * directionX;
        ballY += travelDistanceY * directionY;
        directionY *= -1;
    }
    Console.WriteLine("Collision at {0}/{1}.", ballX, ballY);
}

代码导致以下路径：

Collision at 125/125.
Collision at 225/25.
Collision at 200/0.
Collision at 75/125.
Collision at 0/50.
Collision at 50/0.
Collision at 175/125.
Collision at 225/75.
Collision at 150/0.
Collision at 25/125.
Collision at 0/100.
Collision at 100/0.
Target is located at 225/125.

(225/125)并且（加上球的宽度以获得实际位置）的口袋是B。