Racing With a Two-Speed Bicycle
Math 131 Project 2, Central College, Fall 1999
Due Friday, October 29
A bicycle has two gears, low and high. Performance of the bicycle in each
gear is given below. The performance in low gear is presented in tabular
form, where the table gives the net distance traveled (in feet) against
time (in seconds). For high gear, the performance is represented by a plot
of the bicycle's speed (in feet per second) versus time (in seconds).
Suppose you begin pedaling from a standing start. When should you shift
gears to maximize the distance you will travel in two minutes? Assume that
the switch in gears can be made instantaneously with no loss of speed.
What is the maximum distance you can travel in two minutes?
Be sure you justify your choice of shifting time and your results. A
major portion of your grade will be based on your ability to explain why
no other shifting time will give a greater distance. Be sure you give a
clear explanation of how you calculated the distance you will travel in
two minutes. You may wish to begin by calculating a table of values of
your speed in low gear versus time, or by sketching a graph of speed versus
time.
This project is worth 70 points.
Low Gear Performance
Time
(sec) |
Distance
(feet) |
| 0 |
0.0 |
| 1 |
0.3 |
| 2 |
1.3 |
| 3 |
2.9 |
| 4 |
5.1 |
| 5 |
7.9 |
| 6 |
11.3 |
| 7 |
15.4 |
| 8 |
19.9 |
| 9 |
25.1 |
| 10 |
30.8 |
| 11 |
37.1 |
| 12 |
43.9 |
| 13 |
51.3 |
| 14 |
59.1 |
| 15 |
67.5 |
| 16 |
76.4 |
| 17 |
85.7 |
| 18 |
95.6 |
| 19 |
105.9 |
|
Time
(sec) |
Distance
(feet) |
| 20 |
116.7 |
| 21 |
127.9 |
| 22 |
139.6 |
| 23 |
151.7 |
| 24 |
164.2 |
| 25 |
177.1 |
| 26 |
190.5 |
| 27 |
204.2 |
| 28 |
218.3 |
| 29 |
232.8 |
| 30 |
247.6 |
| 31 |
262.8 |
| 32 |
278.3 |
| 33 |
294.1 |
| 34 |
310.2 |
| 35 |
326.6 |
| 36 |
343.3 |
| 37 |
360.2 |
| 38 |
377.3 |
| 39 |
394.7 |
|
Time
(sec) |
Distance
(feet) |
| 40 |
412.3 |
| 41 |
430.1 |
| 42 |
448.1 |
| 43 |
466.2 |
| 44 |
484.4 |
| 45 |
502.7 |
| 46 |
521.2 |
| 47 |
539.8 |
| 48 |
558.4 |
| 49 |
577.1 |
| 50 |
595.9 |
| 51 |
614.7 |
| 52 |
633.6 |
| 53 |
652.5 |
| 54 |
671.4 |
| 55 |
690.4 |
| 56 |
709.4 |
| 57 |
728.3 |
| 58 |
747.3 |
| 59 |
766.3 |
|
For each second beyond 59, the distance traveled in low gear will increase
by 19.0 feet.
High Gear Performance
Note: A high-resolution plot of the performance in high gear can
be obtained from your instructor.