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# Path Traveled by the Objects and Conservatin of Kinetic Energy

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Author: Jerry Silver

### The Idea

Which path will take the least amount of time for a rolling ball?

• a path that is straight and horizontal, or
• a longer path that starts horizontally, dips in a curved path without excessive friction, and then returns to the same horizontal level it started from.

One path is shorter. So you might think it will take the least amount of time. Because both objects return to the same height, they wind up with the same amount of energy. How does that affect the overall time for the journey? Figure 50- 1 shows the two paths.

### What You Need

• 2 golf balls
• materials to build a track:
• flexible flat wooden molding—one section 8 feet long and one section 6 feet long
• a side board about 6 feet long
• a couple of 2" × 4" × 6" pieces to serve as a base
• small wooden or metal right-angle braces—1 inch corner molding will work
• optional—a basket or plastic cup
• This type of apparatus is also commercially available, as shown in the later Figures 50-6 and 50-7.

### Building the track

1. Draw or sketch the shape of the curved section. This can be traced, copied, or eyeballed. A more exacting approach would be to generate a geometric cycloid and form the curve into that shape.
2. Attach the flexible track to the side board with a straight section, a downward curve returning to a second straight section. Attach the braces to the side board and secure the flexible track to the braces. (Keep the profile of the screws as low as possible, so it does not interfere with the motion of the golf ball. It may be necessary to countersink the screw hole, so the screw head is below the level of the track.)
3. Attach the straight section to the side board, a few inches above the section with the detour.
4. Attach the base in such a way that the path the ball will follow is slightly tilted toward the side board. This minimizes the friction the ball encounters, but it will allow the ball to roll without falling off the track.
5. You may want to add some way to catch the balls after each race to avoid having to chase them every time.
6. A ramp of equal slope and equal length is placed at the start of each path to give objects racing down the two paths the same starting velocity. Be sure to keep any seams in the track as low profile as possible.

### Racing

1. Before doing this, observers can make their prediction. Which track is fastest: a) the flat track b) the track with the detour c) both the same?
2. Release both balls from the same height (above the initial flat section of track) at the same time. Observe the progress of the balls. Repeat a few times to make sure the results are consistent.

### Comparing energy

At the end of the track, regardless of which ball finishes before the other, measure the velocity on the final flat section. You can do this in several ways:

1. Use a motion sensor to measure the speed of the balls on each of the flat sections. If you have two motion sensors, you can measure them at the same time. If you have one, you can do them one after the other. In either case, the most definitive conclusion will result from a good statistical sample.
2. Another way to measure velocity is to take advantage of the fact that the range of a horizontal projectile (as you saw in Project 6) depends only on its height above the ground and the velocity with which it leaves the horizontal surface. In the apparatus shown in Fig 50-1 it is clear that the starting and stopping level for each of the tracks is at a different height above the ground. This does not affect their movement relative to each other. However, it does give the track on top an advantage where it will land unless the height difference is compensated for by raising the landing level. If this is done, balls that move the same distance along the ground have the same velocity. Some designs such as a track used at Michigan State University (http://demo.pa.msu.edu/PicList.asp?DID= DID18) are built with the two tracks side-by-side, so this velocity comparison can be made more easily. Plans for a similar racing-ball track are available from the University of Maryland Physics Department at http:// www.physics.umd.edu/deptinfo/facilities/ lecdem/services/demos/demosc2/c2-11dwg.jpg.
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