Set 1 Problem number 1

• Problem

• Solution

• Generalized Solution

• Explanation in terms of Figure(s), Extension

• Figure(s)

Problem

How far will an object move in 6 seconds if it is moving at a constant speed of 4 meters/second?

Solution

Each second the object moves 4 meters. In 6 seconds it gets to do this 6 times. So it moves a total of 6( 4 meters) = 24 meters.

Generalized Solution

Generalized Response:

If we call the speed v and the time interval `dt, then v represents the number of distance units traveled in a second and `dt represents the number of seconds.

• When we multiply the number of distance units traveled in a second by the number of seconds we get the number of units.
• We therefore say that the distance is the product v `dt. If we let `dist stand for the distance, we have

`dist = v `dt.

Explanation in terms of Figure(s), Extension

The figure below shows how the uniform 1-second distances add up to 2-, 3- and 4-second distances.

• It is easy to visualize how these 1-second distances could add up to 2.5-second, .5-second, 10.7-second or other fractional distances.
• Suggestion: make yourself a sketch of the distances that would correspond to the above time intervals, as well as to intervals of .2, 25, .04 and 6.3 seconds. This will help you firmly fix the idea of distances and time intervals.

The second figure below summarizes certain notation conventions.

• When we use `dt, we really mean 'delta'-t, where 'delta' is the capital Greek letter that looks like a triangle.
• This letter is traditionally associated with a change in a quantity, or the difference between two quantities.
• So `dt is taken to mean 'difference in t'.
• We use the notation `dt instead of using the Greek delta because not all Internet browsers carry the appropriate character sets to represent a delta.

Another important convention is that a 'bar', a short line segment, over the top of a symbol indicates that we are using an average value of that quantity.

• Again this symbol is impossible to represent to a variety of browsers, so we will represent the fact that we are using an average quantity by appending the abbreviation Ave onto the name of the symbol.
• So for example vAve would stand for average velocity.

Figure(s)