Speed is often measured in units of distance per units of time. For example, highway speed limits are often between 50 and 70 miles per hour. The average speed for a period of time is found by dividing:

1. Find the average speed for the following situations.
   1. In the 2000 summer Olympic Games, Australian Cathy Freeman ran the women’s 400-meter race in 49.11 seconds, winning the race.
   2. For a high school track meet, Miguel ran the 100 yard dash in exactly 11 seconds.
   3. How many feet are in 100 yards? Find Miguel’s average speed in feet per second.

   For vacation, Carly and her parents visit Carly’s grandparents, who live 200 miles away. The following graph gives information about one of their trips. The graph shows how far they had driven over time.

   

2. Use the graph of Carly’s trip to complete this table.

   Time Interval	Distance	Time Elapsed	Average Speed
   (hours)	(miles)	(hours)	(mph)
   0-1
   1-1.5
   1.5-2
   2-3.5
   3.5-4.5

3. Rank the time intervals from least steep to steepest. What happens to the average speed as the graph gets steeper?
4. What was Carly’s average speed from 0 hours to 2 hours? From 2 hours to 4 hours?
5. What was Carly’s average speed from 2 hours to 2.5 hours? From 2.5 hours to 3 hours? How can you answer these questions without doing additional calculations?
6. Why couldn’t you answer problem 4 without doing additional calculations? What about Carly’s speed in the intervals in problem 5 is different from her speed in the intervals given in problem 4?
7. Consider the endpoints of each part of the graph described by the time intervals in the table. (That is, look at where the graph is when the time is 0, 1, 1.5, 2, 3.5, and 4 hrs.)
   1. Label each endpoint with its (time, distance) coordinates.
   2. From the coordinates of the endpoints, how can you find the distance traveled over a time interval? How can you find how much time has elapsed?
   3. If the endpoints of a time interval are (a, b) and (x, y), with x > a, how can you find the average speed for that time interval?
8. Suppose you know that the car was moving at a constant speed the whole time from a hours to c hours. Suppose you also know that the distance at time a was b miles, and that the distance at time x was y miles (a < x < c).
   1. What would the graph look like on the interval from a to c hours?
   2. Find the speed from time a to time c.
   3. Find the distance at time c.
9. A measure of the steepness of a line is the slope of the line. The graph on page 5 is actually created from several line segments, each having a different slope. The slope for each is the number value of the speed on the interval. For example, the speed for the segment from 0 to 1 is 30 mph. The slope for that line segment is 30.
   1. Use your answers to previous problems to write a formula for the slope of a line that goes through points (x₁, y₁) and (x₂, y₂).
   2. What is the slope of a horizontal line?

1. 1. about 8.14 m/sec
   2. about 9.09 yd/sec
   3. There are 300 feet in 100 yards; Miguel’s average speed was about 27.27 ft/sec.
2. 
   

   Time Int. (hrs)	Distance (mi)	Time Elapsed (hrs)	Ave. Speed (mph)
   0-1	40	1	40
   1-1.5	30	0.5	60
   1.5-2	0	0.5	0
   2-3.5	90	1.5	60
   3.5-4.5	40	1	40

3. Least steep is 1.5-2 hrs; next steepest are 0-1 and 3.5-4.5 hrs; steepest are 1-1.5 and 2-3.5. As the line gets steeper, the average speed goes up.
4. From 0 to 2 hours, the average speed was 35 mph. From 2 to 4 hours, the average speed was 55 mph.
5. 60 mph in both cases; the average speed of 60 mph didn’t change from 2 to 3.5 hours, so the speeds for intervals within that interval are also 60 mph.
6. For the intervals in problem 4, the car’s speed changed--there were two or more segments with different steepness. The speed didn’t change in the intervals in problem 5.
7. 1. 
      
   2. For distance, subtract the second coordinates; for time elapsed, subtract the first coordinates.
   3. Divide .
8. 1. The graph would be a single line segment--no “bends” or “corners.”
   2. Speed is .
   3. At time c, the distance is b + c().
9. 1. Slope is or, equivalently, .
   2. 0