### Optimum Gear Ratios for a Human-Powered Vehicle

## by Jessica Arce

Submitted : Spring 2012

For this project, I needed to find various values. I needed to find the value for GR_{3}, which is a gear on a human-powered vehicle. I was also required to find the gear ratios GR_{1} and GR_{2} that will minimize the total time needed to reach a velocity of 8 m/s. In addition, I had to calculate the total time needed to reach 8 m/s, the times t_{1}, t_{2}, and t_{3} that the vehicle spends in each gear, and the velocities v_{1} and v_{2} at the shift points from first to second gear and from second to third gear. I accomplished this by integrating the function of dv/dt that was given. I created values for v_{1},v_{2}, and v_{3 }and plugged those into the function of dv/dt that was given and differentiated the function. I then solved for GR_{1 }and GR_{2}. I made a function for the total time needed to reach 8 m/s as well as a function for each individual t_{1}, t_{2}, and t_{3}. I concluded values for GR_{1}, GR_{2}, GR_{3}, the individual times that the vehicle spent in each gear, and the velocities v_{1} and _{v2} at the shift points from first to second gear and from second to third gear.

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