1. INTRODUCTION

Mechanics deals with the motion of various types of bodies, relations of forces, mechanical properties of bodies and matter, etc. Kinematics deals with the motion of particles and rigid bodies without accounting for the forces responsible for their motion. If the body is, however, of small size such that its motion can be described by a point mass moving along a straight line, such a motion is called rectilinear motion or motion in one dimension.

2. MOTION

When the body changes position w.r.t. observer, the body is said to be in motion. If the body doesn’t change position then it is said to be at rest w.r.t. observer. For example, if two passengers are traveling in a moving train they are at rest w.r.t. each other while they are in motion w.r.t. an observer standing on the ground.

3. A PARTICLE

particle is the physical analog of a point. A body of finite size may be considered as a particle only if all parts of the body undergo the same displacement and have the same velocity and acceleration. Thus, for a particle-like body, its motion can be described by studying the motion of any point on the body

4. DISTANCE AND DISPLACEMENT

As a particle moves in space with respect to the time it follows a curve or a straight line, which is called its path or trajectory. Distance is the actual length covered along the path, while displacement is the difference between the final and initial positions of the particle. The position of a particle is given by a position vector drawn from the origin to the particle.

The figure illustrates the difference between distance and displacement. The curve length AB is called the

distance; the vector rAB = rB – rA is called the displacement.

• Distance is a scalar; displacement is
• The magnitude of displacement is always less than or equal to
• For a moving body, displacement can be zero but distance

cannot be zero.

For a body moving in a straight line from point Ato point B along x-axis such that its initial and final position is at a distance xi and xf respectively, the minimum distance between final and initial position is

called displacement, D . x

AB = Dx = x – x .

O                           A             B                                 X

xi                      Dx

f          i                                                                                                                        xf

Displacement is avector frominitialpositionto finalposition. If the pathofthebodyisnot along astraight

line, the total length of the actualpath covered by the body is called the distance covered by thebody and is a scalar quantity. Thedisplacement is independent ofthe actualpathcovered by the body.

5.  AVERAGE SPEED AND AVERAGE VELOCITY

The average speed in a time interval is defined as the total distance travelled by the particle divided by the time interval.

Total distance travelled

Average speed =       Total time taken

The average velocity is defined as

displacement

vav    =   time elapsed

or         v    =

rB - rA = ΔrAB

t B - t A              Δt

• Average speed is a scalar quantity; average velocity is a vector Both are expressed in ms–1or kmh–1.
• Average velocity can be zero but average speed cannot be
• The magnitude of average velocity is always less than or equal to the average

|vav| £ v

or                              £

distance time elapsed

thus        |displacement| £ distance

• Average speed does not mean the magnitude of the average velocity

Consider a body moving on the straight line having the initial position A at time ti at a distance xi from the

origin. It travels a distance x1 to reach B and after travels a distance x2 from B to C reaching at time tf. C is the final position is at a distance xf from the origin. The average speed will be equal to ratio of total

distance covered to the total time.                        x

\ Average speed = x1 + x2

A                         B

Average velocity =

x f - xi t f - ti

O      xi

x _f                                        X

C x

6. INSTANTANEOUS SPEED AND INSTANTANEOUS VELOCITY The instantaneous speed and velocity are the respective average speed and velocityfor the infinitesimally small time interval (e. Dt ® 0). Thus

Instantaneous speed v =

lim

Dr = dr

Dt ®0 Dt     dt

instantaneous velocity v =

lim

Dr = dr

Dt ® 0 Dt dt

• As the time interval tends to zero (Dt ® 0), the displacement vector Dr approaches a limiting direction which is tangential to the path of the particle at that Thus, the instantaneous velocity direction is always tangential to the path followed by the particle.
• The instantaneous speed is equal to the magnitude of the instantaneous
• Aparticle may have constant speed but variable

• For a particle moving with constant velocity, its average velocity and instantaneous velocity are always

Instantaneous velocityof a particlemoving on a straight line say along x-axis is given as

v(t)

lim      D x

=   D t ® 0 D t

lim v .

D t ® 0

The instantaneous velocity v(t) is the derivative of displacement with respect to time.

v (t) =

 d [r(t)] dt