Fastest descent line

1. Content

This is a large exhibition item with three different shaped tracks designed on the booth, one is a straight track and the other two are curved tracks (with different slopes of curvature). Place three small balls at the starting point, open the release mechanism, and make all three balls fall at the same time. At the same time, the corresponding digital tubes will start timing. After all three balls reach the endpoint, the digital tubes will display the time when all three balls have reached the endpoint. We will find that the track with the steepest slope has the shortest drop time for the ball.

The speed at which an object descends along an orbit is not simply determined by the length of the orbit, but also by the shape of the orbit. The greater the component of gravity acting on a sphere along the tangent direction of its orbit, the faster its falling speed. The cycloid trajectory is a circular rolling curve with a steeper trajectory, and the component of gravity in its tangent direction is greater. Therefore, the sphere's falling speed is faster, so it reaches the endpoint first. Although the straight trajectory is shorter, the sphere's falling speed is slower than that of the sphere on the curve, so it reaches the endpoint later.

2. Principle

On a sloping surface, swing two tracks, one is a straight line and the other is a curve, with the same starting and ending heights. Two balls of the same mass and size slide down from the starting point at the same time, while the curved ball reaches the endpoint first. This is because the small ball on the curved track reaches its maximum speed first, so it arrives first. However, there is only one straight line between two points, but there are countless curves. So, which one is the fastest? Galileo proposed this question in 1630, when he believed that this line should be an arc, but later people found that this answer was incorrect. In 1696, Swiss mathematician Johann Bernoulli solved this problem and publicly challenged it to other mathematicians. Newton, Leibniz, Lobida, and Jacob Bernoulli solved this problem. This steepest descent line is a cycloid, also known as a spinning wheel line.

Swiss mathematician John Bernoulli proposed the problem of the steepest descent line again in 1696, seeking a solution. The following year, several mathematicians obtained the correct answer, including members of the Newton, Leibniz, Lobida, and Bernoulli families. The correct answer to this question is the only section of the spiral wheel line that connects two concave points.

The spiral line is the same as the cycloid discussed by Dutch scientist Huygens in 1673. Because the time it takes for a clock pendulum to make a complete swing is equal, the cycloid (spiral wheel line) is also known as the isochronous curve.

3. Purpose

Exhibition items allow the audience to learn about the knowledge of the steepest descent line through interactive participation, while also learning about the application of the steepest descent line through real-life experiences.