Path Traveled by the Objects and Conservation of Kinetic Energy (page 2)
The ball on the flat track will move with constant velocity (Figure 50-2).
The ball following the detour will increase its speed, so it is going fast enough to make up for the extra distance (Figure 50-3).
Once returning to the original height, the ball that went through the detour will return to its original speed. However, now it will be ahead of the ball on the flat track. The ball on the detour will reach the end of the track first (Figure 50-4).
Why It Works
The straight path is easy. The ball travels with the same constant velocity it is given at the start. It does not gain or lose energy, except for the (relatively) small loss due to friction.
On the straight section of the curved path, the second ball travels with the same velocity as the first. As it goes downhill, it picks up speed. If the shape is right, the increase in speed will be more than enough to compensate for the longer distance.
Other Things to Try
In 1696, Johann Bernoulli challenged the most brilliant minds of his day to solve what is now known as the brachistochrone problem—based on the Greek "brichistos" (shortest) and "chronos" (time). Basically, the problem is this: find the path between two points at different levels that an object acted on only by gravity will travel in the least amount of time. This is similar to the racing ball configuration previously defined, except, in this case, the balls start from rest without any initial velocity.
Galileo previously had attempted a solution to this problem. The path Galileo defined was the circular arc connecting the two points. This, although a good approximation, was not the correct solution.
The correct solution was found by five mathematicians who responded to Bernoulli's challenge. This included, among others, a solution by Sir Isaac Newton, which was submitted in just one day. The path taking the shortest time was found to be a mathematical curve known as a cycloid.
A cycloid is defined by the equations x = r(t – sin t) and y = r(1 – cos t), where r can be thought of as the radius of the circle that sweeps out the cycloid and t is time.
An (inverted) cycloid generated (with r = 1, t varying from 0 to 3, and with x values as negative) is shown in Figure 50-5. This is actually similar to a shape generated by a pencil at the edge of circle in a Spirograph.
Although in the racing-ball scenario, we do have a slight head start in the form of an initial velocity, the cycloid curve is a good approximate minimal time path from point A to point B.
An extension to this project would be to build a track that compares a golf ball following a cycloid curve with a straight path down.
Although the large track is more fun to watch, a mini version of either of these tracks can be assembled from foam board with a track shape cut out based on the shape in Figure 50-5 and glued to the baseboard. A clear plastic model can also work and offers the added advantage of working with an overhead projector.
A track system that can be used to study various aspects of conservation of energy can also be purchased. Two examples are shown in Figures 50-6 and 50-7.
The path that takes advantage of increasing the velocity during part of the trip takes less time than one of constant velocity.
This project demonstrates that (aside from frictional losses) energy is conserved. As the potential energy is reduced, it is transformed into kinetic energy. Because both balls have returned to their original height and finished with the same velocity, we confirm that kinetic energy is conserved, regardless of the path traveled.
Warning is hereby given that not all Project Ideas are appropriate for all individuals or in all circumstances. Implementation of any Science Project Idea should be undertaken only in appropriate settings and with appropriate parental or other supervision. Reading and following the safety precautions of all materials used in a project is the sole responsibility of each individual. For further information, consult your state’s handbook of Science Safety.