Position vector in rectilinear motion

Position still requires three coordinates for specification. But, only one of them changes during the motion; remaining two coordinates remain constant. In practice, we choose one dimensional reference line to coincide with the path of the motion. It follows then that position of the particle under motion is equal to the value of x – coordinate (others being zero).

Corresponding position vector also remains a three dimensional quantity. However, if the path of motion coincides with the reference direction and origin of the reference coincides with origin, then position vector is simply equal to component vector in x – direction i.e

The position vectors corresponding to points A, B and C as shown in the figure are 2 i, 4 i and 6 i. units respectively.

Figure 4

Position vectors

As displacement is equal to change in position vector, the displacement for the indicated positions are given as :

AB = ( x 2 - x 1 ) i = Δ x i = ( 4 - 2 ) i = 2 i BC = ( x 2 - x 1 ) i = Δ x i = ( 6 - 4 ) i = 2 i AC = ( x 2 - x 1 ) i = Δ x i = ( 6 - 2 ) i = 4 i AB = ( x 2 - x 1 ) i = Δ x i = ( 4 - 2 ) i = 2 i BC = ( x 2 - x 1 ) i = Δ x i = ( 6 - 4 ) i = 2 i AC = ( x 2 - x 1 ) i = Δ x i = ( 6 - 2 ) i = 4 i

Vector interpretation and equivalent system of scalars

Rectilinear motion involves motion along straight line and thus is described usually in one dimension. Further, rectilinear motion involves only one way of changing direction i.e. the particle under motion can only reverse motion. The particle can move either in positive x-direction or in negative x-direction. There is no other possible direction valid in rectilinear motion.

This attribute of rectilinear motion allows us to do away with the need to use vector notation and vector algebra for quantities with directional attributes like position vector, displacement and velocity. Instead, the vectors are treated simply as scalars with one qualification that vectors in the direction of chosen reference is considered positive and vectors in the opposite direction to the chosen reference is considered negative.

The most important aspect of the sign convention is that a vector like velocity can be expressed by a scalar value say, 5 m/s. Though stated without any aid for specifying direction like using unit vector, the direction of the velocity is indicated, which is in the positive x-direction. If the velocity of motion is -5 m/s, then the velocity is in the direction opposite to the direction of reference.

To illustrate the construct, let us consider a motion of a ball which transverses from O to A to B to C to O along x-axis as shown in the figure.

Figure 5

Motion along straight line

The velocities at various points of motion in m/s (vector form) are :

v O = 2 i v A = 3 i v B = - 4 i v C = 3 i v O = 2 i v A = 3 i v B = - 4 i v C = 3 i

The velocities at various points of motion in m/s (in equivalent scalar form) can be simply stated in scalar values for rectilinear motion as :

v O = 2 v A = 3 v B = - 4 v C = 3 v O = 2 v A = 3 v B = - 4 v C = 3

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 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.

Figure 6

Sing of the tangent of the angle

We may use yet another simpler 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.

Figure 7

Slope of the curve

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 velocity is increasing or decreasing or a constant.

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.

Figure 8

Position – time plot

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.

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).

Figure 9

Velocity – time plot

The representation of the variation of velocity with time, however, needs to be consistent with physical interpretation of motion. For example, we can not think of velocity - time plot, which is a vertical line parallel to the axis of velocity (Figure ii). Such plot is inconsistent as this would mean infinite numbers of values, against the reality of one velocity at a given instant. Similarly, the velocity-time plot should not be intersected by a vertical line twice as it would mean that the particle has more than one velocity (Figure i).

Figure 10

Velocity – time plot

Area under velocity – time plot

The area under the velocity – time plot is equal to displacement. The displacement in the small time period “dt” is given by :

ⅆ x = v ⅆ t ⅆ x = v ⅆ t

Integrating on both sides between time intervals t 1 and t 2 , t 1 and t 2 ,

Δ x = ∫ t 1 t 2 v ⅆ t Δ x = ∫ t 1 t 2 v ⅆ t

The right hand side integral graphically represents an area on a plot drawn between two variables : velocity (v) and time (t). The area is bounded by (i) v-t curve (ii) two time ordinates t 1 t 1 and t 2 t 2 and (iii) time (t) axis as shown by the shaded region on the plot. Thus, the area under v-t plot bounded by the ordinates give the magnitude of displacement (Δx) in the given time interval.

Figure 11

Area under velocity – time plot

When v-t curve consists of negative values of velocity, then the curve extends into fourth quadrant i.e. below time axis. In such cases, it is sometimes easier to evaluate area above and below time axis separately. The area above time axis represents positive displacement, whereas area under time axis represents negative displacement. Finally, areas are added with proper sign to obtain the net displacement during the motion.

To illustrate the working of the process for determining displacement, let us consider the rectilinear motion of a particle represented by the plot shown.

Figure 12

Velocity – time plot

Here,

Area of triangle OAB = 1 2 x 4 x 4 = 8 m Area of trapezium BCDE = - 1 2 x 2 x ( 2 + 4 ) = -6 m Area of triangle EFG = 1 2 x 1 x 2 = 1 m ⇒ Net area = 8 - 6 + 1 = 3 m Area of triangle OAB = 1 2 x 4 x 4 = 8 m Area of trapezium BCDE = - 1 2 x 2 x ( 2 + 4 ) = -6 m Area of triangle EFG = 1 2 x 1 x 2 = 1 m ⇒ Net area = 8 - 6 + 1 = 3 m

A switch from positive to negative value of velocity and vice-versa is associated with change of direction of motion. It means that every intersection of the v – t curve with time axis represents a reversal of direction. Note that it is not the change of slope (from postive to negative and and vive versa) like on the position time that indicates a change in the direction of motion; but the intersection of time axis, which indicates change of direction of velocity. In the motion described in figure above, particle undergoes reversal of direction at two occasions at B and E.

It is also clear from the above example that displacement is given by the net area (considering appropriate positive and negative sign), while distance covered during the motion in the time interval is given by the cumulative area without considering the sign. In the above example, distance covered is :

s = Area of triangle OAB + Area of trapezium BCDE + Area of triangle EFG

⇒ s = 8 + 6 + 1 = 15 m ⇒ s = 8 + 6 + 1 = 15 m

Exercise 1

A person walks with a velocity given by |t – 2| along a straight line. Find the distance and displacement for the motion in the first 4 seconds. What is the average velocity in this period?

Solution 1

Here, the velocity is equal to the modulus of a function in time. It means that velocity is always positive. An inspection of the function reveals that velocity linearly decreases for the first 2 second from 2 m/s to zero. It, then, increases from zero to 2 m/s in the next 2 seconds. In order to obtain distance and displacement, we draw the Velocity – time plot plot as shown.

The area under the plot gives displacement. In this case, however, there is no negative displacement involved. As such, distance and displacement are equal.

Figure 13

Velocity – time plot

Displacement = ΔOAB + ΔBCD Displacement = 1 2 x OB x OA + 1 2 x BD x CD Displacement = 1 2 x 2 x 2 + 1 2 x 2 x 2 = 4 m Displacement = ΔOAB + ΔBCD Displacement = 1 2 x OB x OA + 1 2 x BD x CD Displacement = 1 2 x 2 x 2 + 1 2 x 2 x 2 = 4 m

and the average velocity is given by :

v avg = Displacement Δ t = 4 4 = 1 m/s v avg = Displacement Δ t = 4 4 = 1 m/s

[ Click for Solution 1 ] [ Hide Solution 1 ]

Uniform motion

Uniform motion is a subset of rectilinear motion. It is the most simplified class of motion. In this case, the body under motion moves with constant velocity. It means that the body moves along a straight line without any change of magnitude and direction as velocity is constant. Also, a constant velocity implies that velocity is constant all through out the motion. The velocity at every instant during motion is, therefore, same.

It follows then that instantaneous and averages values of speed and velocity are all equal to a constant value for uniform motion :

v a = | v a | = v = | v | = Constant v a = | v a | = v = | v | = Constant

The motion of uniform linear motion has special significance, as this motion exactly echoes the principle enshrined in the first law of motion. The law states that all bodies in the absence of external force maintain their speed and direction. It follows, therefore, that the study of uniform motion is actually the description of motion, when no external force is in play.

The absence of external force is hypothetical in our experience as bodies are always subject to external force(s). The force of gravitation is short of omnipresent force that can not be overlooked - atleast on earth. Nevertheless, the concept of uniform motion has great theoretical significance as it gives us the reference for the accelerated or the non-uniform real motion.

On the earth, a horizontal motion of a block on a smooth plane approximates uniform motion as shown in the figure. The force of gravity acts and normal reaction force at the contact between surfaces act in vertically, but opposite directions. The two forces balances each other. As a result, there is no net force in the vertical direction. As the surface is smooth, we can also neglect horizontal force due to friction, which could have opposed the motion in horizontal direction. This situation is just an approxmiation for we can not think of a flawless smooth surface in the first place; and also there would be intermolecular attraction between the block and surface at the contact. In brief, we can not achieve zero friction - eventhough the surfaces in contact are perfectly smooth. However, the approximation like this is helpful for it provides us a situation, which is equivalent to the motion of an object without any external force(s).

Figure 14

Uniform motion

We can also imagine a volumetric space, where massive bodies like planets and stars are not nearby. The motion of an object in that space, therefore, would be free from any external force and the motion would be in accordance with the laws of motion. The study of astronauts walking in the space and doing repairs to the spaceship approximates the situation of the absence of external force. The astronaut in the absence of the force of gravitation and friction (as there is no atmosphere) moves along with the velocity of the spaceship.

Motion of separated bodies

The fact that the astronauts moves with the velocity of spaceship is an important statement about the state of motion of the separated bodies. A separated body acquires the velocity of the containing body. A pebble released from a moving train or dropped from a rising balloon is an example of the motion of separated body. The pebble acquires the velocity of the train or the balloon as the case may be.

Figure 15

Motion of separated bodies

It means that we must assign a velocity to the released body, which is equal in magnitude and direction to that of the body from which the released body has separated. The phenomena of imparting velocity to the separated body is a peculiarity with regard to velocity. We shall learn that acceleration (an attribute of non-uniform motion) does not behave in the same fashion. For example, if the train is accelerating at the time, the pebble is dropped, then the pebble would not acquire the acceleration of the containing body.

The reason that pebble does not acquire the acceleration of the train is very simple. We know that acceleration results from application of external force. In this case, the train is accelerated as the engine of the train pulls the compartment (i.e. applies force on the compartment. The pebble, being part of the compartment), is also accelerated till it is held in the hand of the passenger. However, as the pebble is dropped, the connection of the pebble with the rest of the system or with the engine is broken. No force is applied on the pebble in the horizontal direction. As such, pebble after being dropped has no acceleration in the horizontal direction.

In short, we can conclude that a separated body acquires velocity, but not the acceleration.