Resource Lesson
Accelerated Motion: Velocity-Time Graphs
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Whenever an object's velocity changes, the object is said to be
accelerating
. If the acceleration occurs while the object is moving in a straight line, then we say that the object is experiencing
rectilinear acceleration
. An example of this type of acceleration occurs whenever an automaker brags that his vehicle can go from 0-60 mph is "x-number" of seconds. He is assuming that you understand that the car is merely gaining speed, not randomly changing speed in a number of different random directions.
When a velocity-time graph lies in the
1st quadrant
, the object is traveling in a
positive direction
. If the line slopes away from the x- or time axis, it is gaining speed; if it slopes towards the x- or time axis, it is losing speed.
gaining speed
+ acceleration
losing speed
- acceleration
constant speed
0 acceleration
If the velocity-time graph lies in the
4th quadrant
, then the object is losing or gaining speed in a
negative direction
.
gaining speed
- acceleration
losing speed
+ acceleration
constant speed
0 acceleration
Notice that graphically, the acceleration is calculated as the slope of each velocity-time graph.
The graph's slope which equals Δy/ Δx can just as easily be expressed as Δv/Δt, or acceleration.
But be careful! Notice that a positive acceleration does NOT always mean that the object is gaining speed. You cannot forget that acceleration is a vector quantity that represents the change in velocity, another vector quantity. Since vectors have two attributes: magnitude and direction, you can use the rules for signed numbers to remember which combinations result in either a positive or a negative acceleration.
gaining speed (+) in a positive (+) direction
+ acceleration
gaining speed (+) in a negative (-) direction
- acceleration
losing speed (-) in a positive (+) direction
- acceleration
losing speed (-) in a negative (-) direction
+ acceleration
The area bounded by the velocity-graph and the nearest x- or time axis tells you the object's displacement during a specified time interval.
As stated before, whenever the graph is in the 1st quadrant, the object is moving in a positive direction and its area represents a positive displacement. Conversely, whenever the graph is in the 4th quadrant, the object is moving in a negative direction and its area represents a negative displacement. During our study, these areas will either be rectangles, triangles, or a combination of triangles and rectangles. You will have the opportunity to practice calculating areas (or displacements) in the following problem.
Let's look at an example to test our understanding of these properties of velocity-time graphs.
Refer to the following information for the next four questions.
During which extended periods of time was he traveling in a positive direction?
0-3
3-5.5
5.5-6.5
6.5-8
8-9
9-13
13-16.5
none of these
During which extended periods of time was he traveling in a negative direction?
0-3
3-5.5
5.5-6.5
6.5-8
8-9
9-13
13-16.5
none of these
During which extended periods of time was he at rest?
0-3
3-5.5
5.5-6.5
6.5-8
8-9
9-13
13-16.5
none of these
During which extended periods of time was he traveling at a constant velocity?
0-3
3-5.5
5.5-6.5
6.5-8
8-9
9-13
13-16.5
Refer to the following information for the next five questions.
During what time interval did he travel the greatest distance?
0-3
3-5.5
5.5-6.5
6.5-8
8-9
9-13
13-16.5
During what time interval did he travel the least non-zero distance?
0-3
3-5.5
5.5-6.5
6.5-8
8-9
9-13
13-16.5
During which time interval(s) did he experience a negative acceleration?
0-3
3-5.5
5.5-6.5
6.5-8
8-9
9-13
13-16.5
During which time interval(s) did he experience a positive acceleration?
0-3
3-5.5
5.5-6.5
6.5-8
8-9
9-13
13-16.5
During which time interval did he experience an acceleration with the greatest magnitude?
0-3
3-5.5
5.5-6.5
6.5-8
8-9
9-13
13-16.5
none of these
Refer to the following information for the next seven questions.
What total distance did he travel in the first 8 seconds?
What total distance did he travel in the last 8.5 seconds?
What was his average speed in the first 8 seconds?
What was his average speed in the last 8.5 seconds?
What was his average speed for the entire 16.5 seconds?
What was his net displacement during the entire 16.5 seconds?
What was his average velocity during the entire 16.5 seconds?
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