How To Draw Velocity Vs. Time Graphs From Parabolic Position Vs Time Graphs

The Meaning of Shape for a p-t Graph

Our study of i-dimensional kinematics has been concerned with the multiple ways past which the motion of objects tin be represented. Such ways include the use of words, the apply of diagrams, the apply of numbers, the use of equations, and the apply of graphs. Lesson three focuses on the apply of position vs. fourth dimension graphs to describe motion. As we will learn, the specific features of the motility of objects are demonstrated by the shape and the gradient of the lines on a position vs. time graph. The first function of this lesson involves a study of the relationship betwixt the shape of a p-t graph and the motion of the object.

Contrasting a Abiding and a Changing Velocity

To begin, consider a car moving with a abiding, rightward (+) velocity - say of +10 m/s.

If the position-fourth dimension data for such a car were graphed, then the resulting graph would look similar the graph at the correct. Note that a motion described as a abiding, positive velocity results in a line of abiding and positive slope when plotted as a position-time graph.

Now consider a automobile moving with a rightward (+), changing velocity - that is, a car that is moving rightward merely speeding up or accelerating.

If the position-time data for such a car were graphed, and so the resulting graph would wait like the graph at the correct. Note that a movement described equally a irresolute, positive velocity results in a line of changing and positive slope when plotted as a position-time graph.

The position vs. time graphs for the two types of motility - constant velocity and irresolute velocity (acceleration) - are depicted as follows.

Constant Velocity Positive Velocity	Positive Velocity Changing Velocity (dispatch)

The Importance of Gradient

The shapes of the position versus time graphs for these ii basic types of motion - constant velocity motion and accelerated movement (i.e., changing velocity) - reveal an important principle. The principle is that the gradient of the line on a position-time graph reveals useful information nearly the velocity of the object. Information technology is frequently said, "As the gradient goes, and so goes the velocity." Whatever characteristics the velocity has, the gradient volition showroom the same (and vice versa). If the velocity is constant, then the slope is constant (i.e., a directly line). If the velocity is irresolute, then the slope is irresolute (i.e., a curved line). If the velocity is positive, then the gradient is positive (i.e., moving upwards and to the correct). This very principle can be extended to any motion conceivable.

Contrasting a Slow and a Fast Move

Consider the graphs beneath equally example applications of this principle concerning the slope of the line on a position versus time graph. The graph on the left is representative of an object that is moving with a positive velocity (every bit denoted by the positive slope), a constant velocity (as denoted by the abiding gradient) and a small velocity (as denoted by the small slope). The graph on the right has similar features - there is a constant, positive velocity (as denoted by the constant, positive gradient). However, the slope of the graph on the right is larger than that on the left. This larger slope is indicative of a larger velocity. The object represented by the graph on the right is traveling faster than the object represented by the graph on the left. The principle of slope can be used to extract relevant motion characteristics from a position vs. time graph. Equally the slope goes, so goes the velocity.

Ho-hum, Rightward(+) Constant Velocity	Fast, Rightward(+) Constant Velocity

Consider the graphs beneath as another application of this principle of slope. The graph on the left is representative of an object that is moving with a negative velocity (every bit denoted by the negative gradient), a constant velocity (every bit denoted by the abiding slope) and a pocket-size velocity (as denoted by the small-scale slope). The graph on the right has like features - there is a abiding, negative velocity (equally denoted by the constant, negative slope). However, the slope of the graph on the right is larger than that on the left. Once more, this larger gradient is indicative of a larger velocity. The object represented by the graph on the right is traveling faster than the object represented by the graph on the left.

Slow, Leftward(-) Constant Velocity	Fast, Leftward(-) Constant Velocity

Representing an Accelerated Motion

As a final awarding of this principle of gradient, consider the ii graphs below. Both graphs show plotted points forming a curved line. Curved lines have changing slope; they may start with a very small slope and brainstorm curving sharply (either upwardly or downwardly) towards a big slope. In either instance, the curved line of irresolute slope is a sign of accelerated motion (i.e., changing velocity). Applying the principle of slope to the graph on the left, i would conclude that the object depicted by the graph is moving with a negative velocity (since the slope is negative ). Furthermore, the object is starting with a small velocity (the gradient starts out with a minor gradient) and finishes with a large velocity (the slope becomes large). That would mean that this object is moving in the negative management and speeding up (the minor velocity turns into a larger velocity). This is an case of negative acceleration - moving in the negative direction and speeding up. The graph on the right also depicts an object with negative velocity (since there is a negative slope). The object begins with a high velocity (the slope is initially big) and finishes with a small velocity (since the gradient becomes smaller). So this object is moving in the negative direction and slowing down. This is an case of positive acceleration.

Negative (-) Velocity Slow to Fast	Leftward (-) Velocity Fast to Dull

The principle of slope is an incredibly useful principle for extracting relevant information about the motion of objects equally described by their position vs. fourth dimension graph. One time y'all've good the principle a few times, information technology becomes a very natural means of analyzing position-time graphs.

Investigate!

The widget below plots the position-time plot for an object with specified characteristics. The top widget plots the motion for an object moving with a constant velocity. The bottom widget plots the move for an accelerating object. Simply enter the specified values and the widget so plots the line with position on the vertical axis and time on the horizontal axis. Be sure to observe the difference betwixt the constant velocity plot and the accelerated move plot.

We Would Like to Advise ...

Sometimes it isn't enough to just read about it. You take to interact with it! And that's exactly what you lot practise when you employ one of The Physics Classroom's Interactives. We would like to suggest that yous combine the reading of this page with the use of our Graph That Motion or our Graphs and Ramps Interactives. Each is plant in the Physics Interactives section of our website and allows a learner to apply concepts of kinematic graphs (both position-fourth dimension and velocity-fourth dimension) to describe the motion of objects. 

Check Your Understanding

Use the principle of slope to describe the movement of the objects depicted by the 2 plots beneath. In your description, be sure to include such data every bit the direction of the velocity vector (i.eastward., positive or negative), whether there is a constant velocity or an dispatch, and whether the object is moving deadening, fast, from slow to fast or from fast to slow. Be consummate in your description.

Source: https://www.physicsclassroom.com/class/1DKin/Lesson-3/The-Meaning-of-Shape-for-a-p-t-Graph

Posted by: lesherporwhou.blogspot.com

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