# 1.15 Rectilinear motion  (Page 4/6)

 Page 4 / 6

Let us consider the plot generated in the example at the beginning of this module. The data set of the plot is as given here :

```t(s) x(m) Δx(m) --------------------- 0.0 1.0 0.1 0.72 -0.28 0.2 0.48 -0.24 0.3 0.28 -0.20 0.4 0.12 -0.16 0.5 0.00 -0.12 0.6 -0.08 -0.08 0.7 -0.12 -0.12 0.8 -0.12 0.00 0.9 -0.08 0.04 1.0 0.00 0.08 1.1 0.12 0.12 1.2 0.28 0.16 ---------------------```

In the beginning of the motion starting from t = 0, we see that particle covers distance in decreasing magnitude in the negative x - direction. The magnitude of difference, Δx, in equal time interval decreases with the progress of time. Accordingly, the curve becomes flatter. This is reflected by the fact that the slope of the tangent becomes gentler till it becomes horizontal at t = 0.75 s. Beyond t = 0.75 s, the velocity is directed in the positive x – direction. We can see that particle covers more and more distances as the time progresses. It means that the velocity of the particle increases with time and the curve gets steeper with the passage of time.

In general, we can conclude that a gentle slope indicates smaller velocity and a steeper slope indicates a larger velocity.

## Velocity - time plot

In general, the velocity is a three dimensional vector quantity. A velocity – time would, therefore, require additional dimension. Hence, it is not possible to draw velocity – time plot on a three dimensional coordinate system. Two dimensional velocity - time plot is possible, but its drawing is complex.

One dimensional motion, having only two directions – along positive or negative direction of axis, allows plotting velocity – time graph. The velocity is treated simply as scalar speed with one qualification that velocity in the direction of chosen reference is considered positive and velocity in the opposite direction to chosen reference is considered negative.

In "speed- time" and "velocity – time" plots use the symbol “v” to represent both speed and velocity. This is likely to create some confusion as these quantities are essentially different. We use the scalar symbol “v” to represent velocity as a special case for rectilinear motion, because scalar value of velocity with appropriate sign gives the direction of motion as well. Therefore, the scalar representation of velocity is consistent with the requirement of representing both magnitude and direction. Though current writings on the subject allows duplication of symbol, we must, however, be aware of the difference between two types of plot. The speed (v) is always positive in speed – time plot and drawn in the first quadrant of the coordinate system. On the other hand, velocity (v) may be positive or negative in velocity – time plot and drawn in first and fourth quadrants of the two dimensional coordinate system.

## The nature of velocity – time plot

Velocity – time plot for rectilinear motion is a curve (Figure i). The nature of the curve is determined by the nature of motion. If the particle moves with constant velocity, then the plot is a straight line parallel to the time axis (Figure ii). On the other hand, if the velocity changes with respect to time at uniform rate, then the plot is a straight line (Figure iii).

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