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Position - time plot

We use different plots to describe rectilinear motion. Position-time plot is one of them. Position of the point object in motion is drawn against time. Evidently, it is a two dimensional plot. The position is plotted with appropriate sign as described earlier.

Nature of slope

One of the important tool used to understand nature of such plots (as drawn above) is the slope of the tangent drawn on the plot.In particular, we need to qualitatively ascertain whether the slope is positive or negative. In this section, we seek to find out the ways to determine the nature of slope. Mathematically, the slope of a straight line is numerically equal to trigonometric tangent of the angle that the line makes with x – axis. It follows, therefore, that the slope of the straight line may be positive or negative depending on the angle. It is seen as shown in the figure below, the tangent of the angle in first and third quarters is positive, whereas it is negative in the remaining second and fourth quarter. This assessment of the slope of the position - time plot helps us to identify whether velocity is positive or negative?

Sign of the tangent of the angle

We may, however, use yet another simpler and effective technique to judge the nature of the slope. This employs physical interpretation of the plot. We know that the tangent of the angle is equal to the ratio of x (position) and t (time). In order to judge the nature of slope, we progress with the time and determine whether “x” increases or decreases. The increase in “x” corresponds to positive slope and a decrease, on the other hand, corresponds to negative slope. This assessment helps us to quickly identify whether velocity is positive or negative?

Slope of the curve

Direction of motion

The visual representation of the curve might suggest that the tangent to the position – time plot gives the direction of velocity. It is not true. It is contradictory to the assumption of the one dimensional motion. Motion is either in positive or negative x – direction and not in any other direction as would be suggested by the direction of tangent at various points. As a matter of fact, the curve of the position – time plot is not the representation of the path of motion. The path of the motion is simply a straight line. This distinction should always be kept in mind.

In reality, the nature of slope indicates the sense of direction, which can assume either of the two possible directions. A positive slope of the curve denotes motion along the positive direction of the referred axis, whereas negative slope indicates reversal of the direction of motion.

In the position – time plot as shown in the example at the beginning of the module (See Figure) , the slope of the curve from t = 0 s to t = 0.75 s is negative, whereas slope becomes positive for t>0.75 s. Clearly, an inversion of slope indicates reversal of direction. The particle, in the instant case, changes direction once at t = 0.75 s during the motion.

Variation in the velocity

In addtion to the sense of direction, the position - time plot allows us to determine the magnitude of velocity i.e. speed, which is equal to the magnitude of the slope. Here we shall see that the position – time plot is not only helpful in determining magnitude and direction of the velocity, but also in determining whether speed is increasing or decreasing or a constant.

Questions & Answers

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List the application of projectile
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A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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