Student Page 2.4.3 Hopping Hare
Extend what you learned through Trodding Tortoise to analyze the Hopping Hare's motion, data, and graph and relate these to equations that model each part of his race.
1.
The fox made the following observations about the hare's movement during the race with the tortoise:
The hare traveled at a constant rate of 250 meters per minute for the first 2 minutes.
The hare's nap started at exactly 2 minutes and lasted exactly 47.5 minutes.
The hare woke suddenly and traveled at a constant rate of 500 meters per minute during the last minute of the race.
(a)
Complete Table 2.4.3.1 indicating the distance traveled by the hare in the elapsed time.
| Time Interval | Amount of time elapsed |
Change in distance during the time interval |
The hare's speed |
|---|---|---|---|
| 0 to 2 minutes | 250 meters per minute | ||
| 2 to 49.5 minutes | |||
| 49.5 to 50.5 minutes | 500 meters per minute |
(b)
Use the axes you used to draw the tortoise's progress (Student Page Exercise 2.4.2.2) to draw an accurate graph to represent the hare's progress during the race from the information in the table and the fox's observations. What points will help you draw an accurate graph?
(c)
Draw an accurate graph for the hare's progress during the race.
2.
Write an equation relating elapsed time, \(t\text{,}\) and distance, \(D\text{,}\) traveled for the hare's first 2 minutes of travel.
3.
Use the equation you found in Student Page Exercise 2.4.3.2 to answer the following questions:
(a)
How far did the hare travel during the first 15 seconds (0.25 minutes) of the race?
(b)
If the hare had continued traveling at this rate, how long would it have taken him to complete the 1000-meter race? Would the hare have won?
(c)
How far could the hare have traveled if he continued at this rate for the entire 50.5 minutes?
4.
(a)
Find an equation relating the hare's elapsed time and distance traveled for 2 minutes to 49.5 minutes. Keep in mind that he is not sitting on the starting line at the beginning of this time interval.
(b)
What is the slope of the equation you found in Task 2.4.3.4.a? Why does this slope make sense?
5.
(a)
What is the slope of the line that shows the hare's movement during the last minute of the race?
(b)
Is the slope enough to determine an equation for the last minute of the race? Why or why not?
6.
Desmos allows you to plot equations based on time intervals.
(a)
Open a Desmos worksheet. Click on the “?” icon (Help) in the top right corner of the page. Click on the icon for Restrictions and learn how to use them.
(b)
Plot the equation for the tortoise.
(c)
Plot the two equations you found for the hare. For the hare's equations, restrict x to the intervals over which the fox observed him. What do you notice about how Desmos plots the hare's graph?
(d)
(Optional) You already know the slope for the last minute of the hare's travels. Play with a third equation for the final time interval of the hare's race to approximate the equation for this part of the hare's race.