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spacer image The Meaning of Shape for a v-t Graph
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Lesson 4: Describing Motion with Velocity vs. Time Graphs


The Meaning of Shape for a v-t Graph

The study of 1-dimensional kinematics has been concerned with the multiple means by which the motion of objects can be represented. Such means include the use of words, the use of diagrams, the use of numbers, the use of equations, and the use of graphs. Lesson 4 focuses on the use of velocity vs. time graphs to describe motion. The specific features of the motion of objects are demonstrated by the shape and the slope of the lines on a velocity vs. time graph. The first part of this lesson involves a study of the relationship between the motion of an object and the shape of its v-t graph.

Consider a car moving with a constant, rightward (+) velocity of +10 m/s. As you learned in Lesson 1, a car moving with a constant velocity is a car moving with zero acceleration.

car in motion

graphIf the velocity-time data for such a car were graphed, the resulting graph would look like the graph at the right. Note that a motion with constant, positive velocity results in a line of zero slope (a horizontal line has zero slope) when plotted as a velocity-time graph. Furthermore, only positive velocity values are plotted, corresponding to a motion with positive velocity.

Now consider a car moving with a rightward (+), changing velocity – that is, a car that is moving rightward and speeding up or accelerating. Since the car is moving in the positive direction and speeding up, it is said to have a positive acceleration.

car in motion

graphIf the velocity-time data for such a car were graphed, the resulting graph would look like the graph at the right. Note that a motion with changing, positive velocity results in a diagonal line when plotted as a velocity-time graph. The slope of this line is positive, corresponding to the positive acceleration. In addition, only positive velocity values are plotted, corresponding to a motion with positive velocity.

The velocity vs. time graphs for the two types of motion – constant velocity and changing velocity (acceleration) – can be summarized as follows:

Positive Velocity Zero Acceleration Positive Velocity Positive Acceleration
graph graph


The Principle of Slope for a v-t Graph


man

The shapes of the velocity vs. time graphs for these two basic types of motion – constant velocity motion and changing velocity motion (i.e., accelerated motion) – reveal an important principle.

The principle is that the slope of the line on a velocity-time graph reveals useful information about the acceleration of the object. Whatever characteristics the acceleration has, the slope will exhibit the same (and vice versa).

If the acceleration is zero, then the slope is zero (i.e., a horizontal line). If the acceleration is positive, then the slope is positive (i.e., an upward sloping line). If the acceleration is negative, then the slope is negative (i.e., a downward sloping line). This principle can be extended to any motion conceivable.

The slope of a velocity-time graph reveals information about the object's acceleration. But how can you tell if the object is moving in the positive direction (i.e., positive velocity) or in the negative direction (i.e., negative velocity)? And how can you tell if the object is speeding up or slowing down? The answers to these questions hinge on your ability to read a graph.


Positive Velocity vs. Negative Velocity


Since the graph is a velocity-time graph, the velocity is positive whenever the line lies in the positive region (positive y-values, i.e. above the x-axis) of the graph. Similarly, the velocity is negative whenever the line lies in the negative region (negative y-values, i.e. below the x-axis) of the graph. As you learned in Lesson 1, a positive velocity means the object is moving in the positive direction; and a negative velocity means the object is moving in the negative direction. So if an object is moving in the positive direction, the line is located in the positive region of the velocity-time graph (regardless if it is sloping up or sloping down). Likewise, an object is moving in the negative direction if the line is located in the negative region of the velocity-time graph (regardless if it is sloping up or sloping down). Finally, if a line crosses the x-axis from the positive region to the negative region of the graph (or vice versa), then the object has changed directions.

graph


Acceleration vs. Deceleration


How can you tell if the object is speeding up (acceleration) or slowing down (deceleration)? Speeding up means that the magnitude (the value) of the velocity is increasing. For instance, an object with a velocity changing from +3 m/s to + 9 m/s is speeding up. Similarly, an object with a velocity changing from -3 m/s to -9 m/s is also speeding up. In each case, the magnitude of the velocity (the number itself, not the sign or direction) is increasing; the speed is getting larger.

Given this fact, an object is speeding up if the line on a velocity-time graph is changing from a location near the 0-velocity point to a location further away from the 0-velocity point. That is, if the line is moving away from the x-axis (the 0-velocity point), then the object is speeding up. Conversely, if the line is moving towards the x-axis, the object is slowing down.

graph

Link to Animation

See Animations of Various Motions with Accompanying Graphs

Check Your Understanding

graph1. Consider the graph at the right. The object whose motion is represented by this graph is ... (include all that are true):

a. moving in the positive direction.

b. moving with a constant velocity.

c. moving with a negative velocity.

d. slowing down.

e. changing directions.

f. speeding up.

g. moving with a positive acceleration.

h. moving with a constant acceleration.


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Capturé par MemoWeb ŕ partir de http://www.physicsclassroom.com/Class/1DKin/U1L4a.html  le 17/11/01