The Tortoise and the Hare

This is a calculus question that required some creativity in regards to Rate of Change in a race.

The question was as follows:

Situation

This discussion will center around the following situation. Torty and Harry are competing in a 100m footrace. Torty’s average speed on any 5-second interval is always less than Harry’s average speed on the same 5-second interval, but Torty wins the race!

Additional Details

The race unfolds much like a typical footrace would. Both racers start at 𝑡=0 in the same position, run directly forward with a positive velocity on a straight, flat track with no obstacles, and the first person to run a total of 100 meters will win the race. We require the race to last at least 10 seconds, but you can make the race last longer if you wish. For ease of measurement, it will be beneficial to consider Torty and Harry to keep running forward after 100 meters, so their speeds can always be calculated by looking forward in time even though the race ends at 100 meters.

Requirements

Discuss (using the concepts of constant and average rates of change) how it is possible that Torty wins the race.

Post an initial conjecture that describes the properties of Torty and Harry’s distance-time relationships that would allow Torty to win the race despite the stated average rate of change (AROC) constraints.

Post a graph that visually depicts your possible distance-time curves for Torty and Harry that satisfy the race constraints. Include (in words) a description of why the curves satisfy the AROC requirements of the race. Also, include graphical depictions of different 5-second AROCs for Torty and Harry that further suggest the ways in which the curves meet the requirements. Note: At least one of the curves in your final solution must be nowhere linear (i.e., there is no interval over which the function’s graph is a line); however, these brainstorming curves can be whatever you want.

Upload a final, neatly drawn (or computer-generated) graph that shows both Torty and Harry’s distance-time relationships satisfying the race constraints. At least one of the curves must be nowhere linear (i.e., there is no interval over which the function’s graph is a line). You will each need to include a description (in words) of why the graphs visualize the constraints of the race and a few drawn examples of Torty and Harry’s AROCs that show the constraints of the race being met.