Motion in a straight line || 1-D motion || Rectilinear motion || Straight path motion || Chapter 3 Physics || Notes ||

                      1.  Introduction

 Motion is change in position of an object with time. In this chapter, we shall learn how to describe motion. For this, we develop the

concepts of velocity and acceleration. We shall confine ourselves to the study of motion of objects along a straight line, also known as rectilinear motion. For the case of rectilinear motion with uniform acceleration, a set of simple equations can be obtained. Finally, to understand the relative nature of motion, we introduce the concept of relative velocity.

In our discussions, we shall treat the objects in motion as point objects. This approximation is valid so far as the size of the object is much smaller than the distance it moves in a

reasonable duration of time. In a good number of situations in real-life, the size of objects can be neglected and they can be considered as point-like objects without much error.

        2. Path length  and  Displacement

2.1 Path length

We choose the x-axis such that it coincides with the path of the car’s motion and origin of

the axis as the point from where the car started moving, i.e. the car was at x = 0 at t = 0 .

Let P, Q and R represent the positions of the car at different instants of time. Consider two cases of motion. In the first case, the car moves from O to P. Then the distance moved by the car is OP = +360 m. This distance is called the path length traversed by the car. In the second case, the car moves from O to P and then moves back from P to Q. During this course of motion, the path length traversed is OP + PQ = + 360 m + (+120 m) = + 480 m. Path length is a scalar quantity — a quantity that has a magnitude only and no direction. 

         

2.2 Displacement

Let x1 and x2 be the positions of an object at time t1 and t2 . Then its displacement,denoted by ∆x, in time ∆t = (t2 - t1), is given by the difference between the final and initial positions : 

 ∆x = x2 – x1

(We use the Greek letter delta (∆) to denote a

change in a quantity.)

If x2  > x1 , ∆x is positive; and if x2 < x1 , ∆x is 

negative.

Displacement has both magnitude and

direction. Such quantities are represented by

vectors. You will read about vectors in the next  chapter. Presently, we are dealing with motion along a straight line (also called rectilinear motion) only. In one-dimensional motion, there are only two directions (backward and forward, upward and downward) in which an object can 

move, and these two directions can easily be

specified by + and – signs.

Displacement is the shortest path travelled by an object in a particular period of time. The displacement has a magnitude of 360 m and

is directed in the positive x direction as indicated  by the + sign. Similarly, the displacement of the car from P to Q is 240 m – 360 m = – 120 m. The negative sign indicates the direction of

displacement. Thus, it is not necessary to use

vector notation for discussing motion of objects in one-dimension.

The magnitude of displacement may or may

not be equal to the path length traversed by

an object. For example, for motion of the car

from O to P, the path length is +360 m and the

displacement is +360 m. In this case, the

magnitude of displacement (360 m) is equal to

the path length (360 m). But consider the motion 

of the car from O to P and back to Q. In this

case, the path length = (+360 m) + (+120 m) = +

480 m. However, the displacement = (+240 m) –

(0 m) = + 240 m. Thus, the magnitude of

displacement (240 m) is not equal to the path

length (480 m).

The magnitude of the displacement for a

course of motion may be zero but the

corresponding path length is not zero. 

If an object moving along the straight line

covers equal distances in equal intervals of

time, it is said to be in uniform motion along a

straight line.   

    3. Average speed and Average velocity

Average velocity as defined above involves

only the displacement of the object. We have seen  earlier that the magnitude of displacement may be different from the actual path length. To describe the rate of motion over the actual path, we introduce another quantity called average 

speed.

Average speed is defined as the total path

length travelled divided by the total time

interval during which the motion has taken

place :

Average speed= Total path length /

Total time interval

Average velocity is defined as the

change in position or displacement (∆x) divided  by the time intervals (∆t), in which the displacement occurs : 

 

v=x2-x1/t2-t1 =  ∆x/∆t

where x2 and x1 are the positions of the object at time t2and t1, respectively. Here the bar over the symbol for velocity is a standard notation used to indicate an average quantity. The SI unit for velocity is m/s or m s–1, although km h–1 is used in many everyday applications. 

      4.Instantaneous velocity and speed 

The average velocity tells us how fast an object  has been moving over a given time interval but does not tell us how fast it moves at different instants of time during that interval. For this, we define instantaneous velocity or simply velocity v at an instant t. 

The velocity at an instant is defined as the

limit of the average velocity as the time interval  ∆t becomes infinitesimally small. 

Instantaneous speed or simply speed is the

magnitude of velocity. For example, a velocity of  + 24.0 m s–1 and a velocity of – 24.0 m s–1 — both  have an associated speed of 24.0 m s-1. It should be noted that though average speed over a finite interval of time is greater or equal to the magnitude of the average velocity, 

instantaneous speed at an instant is equal to

the magnitude of the instantaneous velocity at

that instant.

                     5. Acceleration

Acceleration is defined as the rate of change of velocity in per unit time. 

The average acceleration a over a time

interval is defined as the change of velocity

divided by the time interval : velocity in per unit time. 

                  A=v2-v1/t2-t1 =  ∆v/∆t

where v2 and v1 are the instantaneous velocities  or simply velocities at time t2 and t1. 

It is the average change of velocity per unit time. The SI unit of acceleration is m s–2 . 

On a plot of velocity versus time, the average

acceleration is the slope of the straight line

connecting the points corresponding to (v2, t2)

and (v1, t 1).

Note that the x-t, v-t, and a-t graphs shown

in several figures in this chapter have sharp

kinks at some points implying that the

functions are not differentiable at these

points. In any realistic situation, the

functions will be differentiable at all points

and the graphs will be smooth.

What this means physically is that

acceleration and velocity cannot change

values abruptly at an instant. Changes are

always continuous.

          6. KINEMATIC EQUATIONS FOR

     UNIFORMLY ACCELERATED MOTION

1. First method: In the first method, we split  the path in two parts : the upward motion (A to B) and the downward motion (B to C) and calculate the corresponding time taken t1 and t2. Since the velocity at B is zero, we have: 

 v = vo + at

0 = 20 – 10t1

Or, t1 = 2 s

This is the time in going from A to B. From B, or  the point of the maximum height, the ball falls freely under the acceleration due to gravity. The ball is moving in negative y direction.

2. Second method: The total time taken can  also be calculated by noting the coordinates of  initial and final positions of the ball with respect to the origin chosen. 

Note that the second method is better since we  do not have to worry about the path of the motion as the motion is under constant acceleration.

         

                7. Relative Velocity

You must be familiar with the experience of

travelling in a train and being overtaken by

another train moving in the same direction as

you are. While that train must be travelling faster  than you to be able to pass you, it does seem slower to you than it would be to someone standing on the ground and watching both the trains. In case both the trains have the same velocity with respect to the ground, then to you the other train would seem to be not moving at all. To understand such observations, we now 

introduce the concept of relative velocity. 

WORK, POWER and ENERGY

LAWS OF MOTION

MOTION IN A PLANE

HIMANSHU CHAUDHARY