Calculus in Longboarding
Kyle Lulewicz
Honors Contract
November 25, 2012
Kathleen Brewer
MAT 265
Introduction
Within the sport of longboarding, there is a large argument as far as which wheels perform the best when ridden. Some will say a wheel with a larger diameter will perform better because it will take fewer revolutions to generate the same speed as a wheel with a smaller diameter. Others say the smaller diameter will perform better because the contact patch (the part of the wheel that is in contact with the ground) is smaller resulting in less friction while riding and performing slides. In downhill riding, a slide is done to shed off speed before going into a turn. Other uses for slides are strictly for fun and style. Whichever diameter wheel can shed off the most speed in a specific distance would essentially perform better for a downhill run with sharp turns.
Experiment
To see which type of wheel had an overall better performance, three tests were done with both 70mm and 83mm wheels. These tests included a straight downhill run, a downhill run with a slide in the middle to duplicate the need to reduce speed, and a shutdown slide. The wheels that were chosen have the same durometer rating, meaning the density of the urethane of the wheel is the same. This means that the difficulty to initiate a slide was the same on both wheels. The same hill, rider, and board were used for each test, leaving the diameter of the wheels as the only variable.
Lines were drawn on the pavement every 25ft for approximately 250ft. Several test runs were done before collecting data to see if the rider could maintain a consistent ride for every trial. After the test runs were completed, the rider then proceeded with the various tests. Upon reaching each line on the pavement, a stopwatch was used to record the time it took to reach it. After all six tests (three for each type of wheel), the data was recorded in an excel spreadsheet. The position vs. time data was turned into velocity vs. time and acceleration vs. time by taking the derivatives. Since an accurate function could not be computed on a calculator, the derivatives were done by taking the slopes at specific points and graphing them. After doing such calculations, the results were very conclusive.
Results
70mm Shutdown Slide Distance = 27.5ft
83mm Shutdown Slide Distance = 32ft
(for position, velocity, and acceleration results for other two tests,
see attached spreadsheets)
Discussion
According to the results, neither wheel can be claimed as the better performer. The downhill tests first showed that the diameter has little effect over speed. Although the 83mm wheel finished .25 seconds faster than its opponent, the 70mm had a higher top speed than the 83mm wheel. However, throughout the run, the larger wheel was more consistent at keeping its speed up. The smaller wheel had less consistency, but a faster top speed. During the slide runs, the 70mm wheel had an acceleration of -4.1ft/s/s at the slide whereas the 83mm had -3.18ft/s/s acceleration. Each slide was done at the 100ft mark for a distance of 25ft and the velocity of both tests was within .5ft/s of each other. Both slide runs took approximately 15.9 seconds to complete resulting in the 70mm wheel to be the better as far as reducing speed in a specific distance. For the final test, a shutdown slide was done at 150ft to determine which diameter took longer to stop. The 70mm wheel took approximately 27.5ft to stop and the 83mm one took 32ft. The velocity of the board was slightly faster during the 70mm run resulting in the smaller wheel being better for stopping.
Conclusion
Overall, the results of the tests show that neither wheel has an overall better performance than its opponent. As far as their individual specialties go, the larger wheel was much more consistent and was faster than the 70mm wheel. The smaller wheel however, actually slowed down faster than the larger wheel, therefore debunking the idea that there is less friction involved. Human error could have played a role in the results, but not enough to make any conclusions. The final results can conclude that the diameter of the wheel does not make a very substantial difference in the overall performance of the board.