Learning Resources

Lesson

You have experienced the word rate used in various contexts. For example,  if you have a part time or summer job you receive a certain rate of pay. If you visit a doctor (s)he will likely take your heart rate. If you borrow money you pay a certain rate of interest.

In this lesson we will deal with rate of change. This is defined as the change in one quantity compared with the change in another quantity. The mathematical notation for change is D , so if one quantity is represented by x and another quantity represented by y, then the rate of change can be expressed by the formula:

One rate of change that we are all familiar with is speed. Speed is the change in distance compare to the change in time. Those of you doing physics constantly refer to speed by its more correct term of rate and use the formulas:

distance = rate x time       or       

Using the notation above, we could also write:  

Before you begin Investigation 1, be aware of the different units that can be used to measure speed. In the examples given, the time unit varies between minutes and hours. The examples below show how to change from km/hr to km/min and from km/min to km/hr.

Essentially, in the first case you divide by 60 and in the second case you multiply by 60.

We will first look at the concept of average speed, which is the speed you have to maintain in order to travel a specific distance in a specific amount of time. Average speed does not tell you how fast you are traveling at a particular point in time. This is illustrated in the following example:

Number of minutes from start	Total number of kilometres traveled
10	10
30	40

The total distance traveled for the entire trip was 40 km and the total time to cover that distance was 30 min. We can calculate the average speed of the entire trip as follows:

average speed = 40 km / 30 min = km/min =  60 x km/hr = 80 km/hr.

However, if we look at the second part of the trip only, we notice that here he traveled a distance of 30 km (40 - 10 the change in distance or Dd) in a time of 20 min (30 - 10 the change in time or Dt). If we calculate the average speed for this part of the trip we get:

average speed = 30 km / 20 min = 1.5 km/min = 60 x 1.5 km/hr = 90km/hr.

So even though the average speed for the entire trip was 80 km/hr, the speed on the second part actually exceeded this. Similarly, during that second part we do not know if a constant speed was maintained. All we know is the average speed for that section.

This should give you enough background to be able to complete Investigation 1 on your own.

Activity

1. Complete Investigation 1 on page 76 in your text.
2. Complete the Investigation Questions 1 - 5 on page 77.
3. Do the CYU Question 6 on page 77.

When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. The is a good graph of the problem on Page 144 of the TRB that you might want to photocopy and put in your binder for future reference. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.

After you do the assigned activities, click on the Test Yourself button at the top of the page for a quick quiz on this  lesson.

Test Yourself

Use the information in the table below to answer the questions which follow it. 

Number of minutes from start of trip	Total number of kilometres traveled
40	50
60	70
90	110

1. Predict how the average speed is changing.
2. Calculate the average speed for each of the three time periods.
3. Calculate the average speed for the entire trip.
4. Discuss why the overall average speed makes sense with respect to the three partial-trip speeds.

Solution