CHAPTER TWO
MOTION IN A STRAIGHT LINE
2.1 INTRODUCTION
Motion is common to everything in the universe. We walk,
run and ride a bicycle. Even when we are sleeping, air moves
2.1 Introduction into and out of our lungs and blood flows in arteries and
2.2 Instantaneous velocity and veins. We see leaves falling from trees and water flowing
speed down a dam. Automobiles and planes carry people from one
2.3 Acceleration place to the other. The earth rotates once every twenty-four
2.4 Kinematic equations for hours and revolves round the sun once in a year. The sun
uniformly accelerated motion itself is in motion in the Milky Way, which is again moving
2.5 Relative velocity within its local group of galaxies.
Summary
Motion is change in position of an object with time. How
Points to ponder does the position change with time ? In this chapter, we shall
Exercises 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.
In Kinematics, we study ways to describe motion without
going into the causes of motion. What causes motion
described in this chapter and the next chapter forms the
subject matter of Chapter 4.
2024-25
, 14 PHYSICS
2.2 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. In other words,
∆x
v = lim (2.1a)
∆t → 0 ∆t
Fig. 2.1 Determining velocity from position-time
dx (2.1b) graph. Velocity at t = 4 s is the slope of the
=
dt tangent to the graph at that instant.
lim
where the symbol ∆t →0 stands for the operation Now, we decrease the value of ∆t from 2 s to 1
of taking limit as ∆tg0 of the quantity on its s. Then line P1P2 becomes Q1Q2 and its slope
right. In the language of calculus, the quantity gives the value of the average velocity over
on the right hand side of Eq. (2.1a) is the the interval 3.5 s to 4.5 s. In the limit ∆t → 0,
differential coefficient of x with respect to t and the line P1P2 becomes tangent to the position-
dx time curve at the point P and the velocity at t
is denoted by (see Appendix 2.1). It is the
dt = 4 s is given by the slope of the tangent at
rate of change of position with respect to time, that point. It is difficult to show this
at that instant. process graphically. But if we use
numerical method to obtain the value of
We can use Eq. (2.1a) for obtaining the
the velocity, the meaning of the limiting
value of velocity at an instant either
process becomes clear. For the graph shown
graphically or numerically. Suppose that we
in Fig. 2.1, x = 0.08 t3. Table 2.1 gives the
want to obtain graphically the value of
value of ∆x/∆t calculated for ∆t equal to 2.0 s,
velocity at time t = 4 s (point P) for the motion
1.0 s, 0.5 s, 0.1 s and 0.01 s centred at t =
of the car represented in Fig.2.1 calculation.
4.0 s. The second and third columns give the
Let us take ∆t = 2 s centred at t = 4 s. Then,
by the definition of the average velocity, the ∆t ∆t
value of t1= t − and t 2 = t + and the
slope of line P1P2 ( Fig. 2.1) gives the value of 2 2
average velocity over the interval 3 s to 5 s. fourth and the fifth columns give the
∆x
Table 2.1 Limiting value of at t = 4 s
∆t
2024-25
MOTION IN A STRAIGHT LINE
2.1 INTRODUCTION
Motion is common to everything in the universe. We walk,
run and ride a bicycle. Even when we are sleeping, air moves
2.1 Introduction into and out of our lungs and blood flows in arteries and
2.2 Instantaneous velocity and veins. We see leaves falling from trees and water flowing
speed down a dam. Automobiles and planes carry people from one
2.3 Acceleration place to the other. The earth rotates once every twenty-four
2.4 Kinematic equations for hours and revolves round the sun once in a year. The sun
uniformly accelerated motion itself is in motion in the Milky Way, which is again moving
2.5 Relative velocity within its local group of galaxies.
Summary
Motion is change in position of an object with time. How
Points to ponder does the position change with time ? In this chapter, we shall
Exercises 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.
In Kinematics, we study ways to describe motion without
going into the causes of motion. What causes motion
described in this chapter and the next chapter forms the
subject matter of Chapter 4.
2024-25
, 14 PHYSICS
2.2 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. In other words,
∆x
v = lim (2.1a)
∆t → 0 ∆t
Fig. 2.1 Determining velocity from position-time
dx (2.1b) graph. Velocity at t = 4 s is the slope of the
=
dt tangent to the graph at that instant.
lim
where the symbol ∆t →0 stands for the operation Now, we decrease the value of ∆t from 2 s to 1
of taking limit as ∆tg0 of the quantity on its s. Then line P1P2 becomes Q1Q2 and its slope
right. In the language of calculus, the quantity gives the value of the average velocity over
on the right hand side of Eq. (2.1a) is the the interval 3.5 s to 4.5 s. In the limit ∆t → 0,
differential coefficient of x with respect to t and the line P1P2 becomes tangent to the position-
dx time curve at the point P and the velocity at t
is denoted by (see Appendix 2.1). It is the
dt = 4 s is given by the slope of the tangent at
rate of change of position with respect to time, that point. It is difficult to show this
at that instant. process graphically. But if we use
numerical method to obtain the value of
We can use Eq. (2.1a) for obtaining the
the velocity, the meaning of the limiting
value of velocity at an instant either
process becomes clear. For the graph shown
graphically or numerically. Suppose that we
in Fig. 2.1, x = 0.08 t3. Table 2.1 gives the
want to obtain graphically the value of
value of ∆x/∆t calculated for ∆t equal to 2.0 s,
velocity at time t = 4 s (point P) for the motion
1.0 s, 0.5 s, 0.1 s and 0.01 s centred at t =
of the car represented in Fig.2.1 calculation.
4.0 s. The second and third columns give the
Let us take ∆t = 2 s centred at t = 4 s. Then,
by the definition of the average velocity, the ∆t ∆t
value of t1= t − and t 2 = t + and the
slope of line P1P2 ( Fig. 2.1) gives the value of 2 2
average velocity over the interval 3 s to 5 s. fourth and the fifth columns give the
∆x
Table 2.1 Limiting value of at t = 4 s
∆t
2024-25
