Name  ________________________________  Date  ______________  Period ______

Graphing Velocity

Overview

      It is important to describe motion in terms of position and time, rather than distance.  Position is much less ambiguous than distance (sometimes regarded as the path length, sometimes as displacement).  Some authors use 's' to describe this variable; we prefer 'x' for horizontal motion (and 'y' when the motion is vertical).  We advise against the use of 'd'.  When it comes time to discuss the slope of the position-time graph, the definition for velocity, ,naturally arises.  Change in position is superior to change in distance; the latter is a difference of differences.  Change in position is the definition of displacement, the quantity that helps distinguish velocity from speed.  Displacement can be (+) or (-), distance is by definition (+).

Graphing

     The shape of a velocity versus time graph reveals pertinent information about an object's acceleration. For example, if the acceleration is zero, then the velocity-time graph is a horizontal line (i.e., the slope is zero). If the acceleration is positive, then the line is an upward sloping line (i.e., the slope is positive). If the acceleration is negative, then the velocity-time graph is a downward sloping line (i.e., the slope is negative). If the acceleration is great, then the line slopes up steeply (i.e., the slope is great).

When discussing the meaning of the graphs, be sure to use a wide variety of examples.

The object starts somewhere to the right of the origin and moves to the left at constant speed.

Example 1:  Constant Velocity

Consider a car moving with a constant velocity of +10 m/s. A car moving with a constant velocity has an acceleration of 0 m/s/s.

The velocity-time data and graph would look like the graph below. Note that the line on the graph is horizontal. That is the slope of the line is 0 m/s/s. In this case, it is obvious that the slope of the line (0 m/s/s) is the same as the acceleration (0 m/s/s) of the car.

So in this case, the slope of the line is equal to the acceleration of the velocity-time graph.

Example 2:  Changing Velocity

Now consider a car moving with a changing velocity. A car with a changing velocity will have an acceleration.

     The velocity-time data for this motion show that the car has an acceleration value of 10 m/s/s. The graph of this velocity-time data would look like the graph below. Note that the line on the graph is diagonal - that is, it has a slope. The slope of the line can be calculated as 10 m/s/s. It is obvious once again that the slope of the line (10 m/s/s) is the same as the acceleration (10 m/s/s) of the car.

      In both instances above, the slope of the line was equal to the acceleration. As a last illustration, we will examine a more complex case. Consider the motion of a car which first travels with a constant velocity (a=0 m/s/s) of 2 m/s for four seconds and then accelerates at a rate of +2 m/s/s for four seconds. That is, in the first four seconds, the car is not changing its velocity (the velocity remains at 2 m/s) and then the car increases its velocity by 2 m/s per second over the next four seconds. The velocity-time data and graph are displayed below. Observe the relationship between the slope of the line during each four-second interval and the corresponding acceleration value.

A motion such as the one above further illustrates the important principle: the slope of the line on a velocity-time graph is equal to the acceleration of the object. This principle can be used for all velocity-time in order to determine the numerical value of the acceleration.

Name _____________________________________  Date _________  Per ________

Discussion Questions:

1. It is important to understand motion in terms of ______________ and ____________.
2. Displacement can be __________ or _________ .
3. Distance is always _______________.
4. What does the shape of the distance-time Graph show us about acceleration?

5. Using the picture below, explain it in terms of the velocity of the car?

6. What does the motion map look like from question # 5?

7. What does a graph of constant velocity look like?

8. Using the picture below, explain it in terms of the velocity of the car?

9. What does the motion map look like for question number 8.

10. What does a graph of different velocities look like?

11. The slope of the line on a velocity-time graph is ______ to the acceleration of the object