INTRODUCTION

Kinematics is that branch of mechanics, which describes the motion of bodies without reference to the forces that either cause the motion or are generated as a result of the motion.  Kinematics is often referred to as the ‘geometry of motion’. We start our study of kinematics by first discussing in this chapter the motion of a particle. A particle is a physical analogue of a point. A particle is a body whose physical dimensions are so small compared with the radius of curvature of its path that we can treat the motion of a particle as that of a point. Later on, we shall apply the concepts learnt here to the motion of rigid bodies, which are a collection of particles.

Rest and Motion

An object is said to be in motion if its position changes with respect to its surroundings in a given time. On the other hand, if the position of the object does not change with respect to its surroundings, it is said to be at rest. A car speeding on the road, a ship sailing on water and a bird flying through the air are examples of objects in motion. A book lying on a desk is at rest because its position with respect to the desk does not change with time.

If a person sitting in a boat is crossing a river, then the person with respect to the boat is in a state of rest (because his position with respect to the boat is not changing), but with respect to the shore he is in the state of motion. Similarly, if two cars are going side by side with same speed then with respect to each other they are in a state of rest, but with respect to trees and persons on the road they are in a state of motion. Thus, it is clear that description of motion depends on the observer or what is called in the language of Physics as a ‘Frame of Reference’. Thus, in the example of a person sitting in a boat and crossing the river, in a frame of reference attached to the boat, the person is at rest, while in a frame of reference attached to the shore the person is in the state of motion.

A convenient way to fix a frame of reference is to choose an origin and three mutually perpendicular axes labelled as x, y and z axes. Then the position of an object in space is, specified by the three coordinates. As the object moves, one or two or all the three coordinates change with time and it is the essential task of mechanics to obtain these coordinates as functions of time. If we know x(t), y(t) and z(t), then the motion of the object is completely described.

The motion of an object is said to be one dimensional when only one of the three coordinates specifying the position of the object changes with time. The motion of a car on a road, the motion of a train along a railway track or an object falling freely are examples of one dimensional motions. One dimensional motion is also termed as rectilinear motion. The motion of an object  is said to be two dimensional when two of the three coordinates specifying the position of the object change with time. The motion of a planet around the sun, a body moving along the circumference of a circle are examples of motion in two dimensions. Two dimensional motion is also referred to as motion in a plane.

The actual path followed by an object in a particular reference frame is termed as its “trajectory”. Thus, the trajectory is a straight line in case of one dimensional motion whereas in case of a two dimensional motion, the trajectory can be a circle, a parabola or in general, a curve.

Note

AIEEE Syllabus has only the description of one and two dimensional motions. Three dimensional motions, wherein all the three coordinates specifying the position of the object change with time, will seldom occur (as for example, the motion of a charged particle when projected in a magnetic field at an angle other than 90^o. Such a particle describes a helix, a three dimensional trajectory).

Bodies that have only motion of translational motion behave like particles. An observer will describe the motion as translational if the axes of a reference frame which is imagined rigidly attached to the object always maintain the same orientation in space with respect to the observer (see figure).

In addition, if these axes change their orientation with respect to the observer, the motion is said to be combined translational and rotational.
We shall learn about the rotational part of the motion in the next unit. Presently, we begin the description of translational motion.

SOME BASIC DEFINITIONS

Distance and Displacement
The position of a moving object changes with respect to time. The length of the actual path covered by a body in a time interval is called distance, while the difference between the final and initial positions of an object is called displacement.

In the figure, let a particle be at point A at time t₁, its position in the x-y plane being described by position vector r1→ . At a later time t_2, let this particle be at point B, described by position vector r1→ . The displacement vector describing the change in position of the particle as it moves from A to B is
Δr→=r2→−r1→ The distance travelled by the particle is the length AB along the curve.

Distance is a scalar quantity which has magnitude only. Displacement is a vector quantity which has both magnitude and direction. Suppose PIE EDUCATION is at a distance of 5 km from your house and you come to PIE EDUCATION and go back to your house. Although you have travelled a distance of 10 km but your displacement is zero (null vector). If a body is moving in a circular path, then after one rotation its displacement will be zero but the distance travelled will be equal to the circumference of the circle. Thus, in general, magnitude of displacement is not equal to the distance travelled. However, it can be so if the motion is along a straight line without change in direction.

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. Thus
Averagespeed=TotaldistancetravelledTotaltimetaken
Average speed is a scalar quantity and its unit is m/s  or  km/h.
The average velocity (see figure) is defined as
<v⃗ >=displacementelapsedtime=r⃗ 2−r⃗ 1t2−t1=Δr⃗ Δt
The average velocity is a vector quantity having the same direction as displacement. Its unit is also meter/second or km/h.

Application 1

A bird flies toward east at 10 m/s for 100 m. It then turns around and flies at 20 m/s for 15 s. Find
(a) its average speed
(b) its average velocity

Solution

Let us take the x axis to point east. A sketch of the path is shown in the figure. To find the required quantities, we need the total time interval. The first part of the journey took
Δt1=(100m)(10m/s)=10s
and we are given Δt2= 15 s for the second part. Hence the total time interval is
Δt=Δt1+Δt2= 25 s
The bird flies 100 m east and then (20 m/s) (15s) = 300 m west
(a) Average speed = DistanceΔt=100m+300m25s=16m/s
(b) The net displacement is = Δx=Δx1+Δx2 = 100m – 300 m = -200 m
So that
vav=ΔxΔt=−200m25s=−8m/s
The negative sign means that vav is directed toward the west.

Application

A jogger runs his first 100 m at 4 m/s and the second 100 m at 2 m/s in the same direction. What is the average velocity ?

Solution

A sketch of his motion is shown in figure. His net displacement
Δx=Δx1+Δx2= 100 m+ 100 m= 200 m
The first half took
Δt1=(100m)(4m/s)=25s
while the second took
Δt2=(100m)(2m/s)=50s
The total time interval is
Δt=Δt1+Δt2= 75 s.
Therefore, his average velocity is
vav=ΔxΔt=200m75s=2.67m/s
Since 2.76≠12(4+2), we see that the average velocity is not, in general, equal to the average of the velocities.

Instantaneous Speed and Instantaneous Velocity

Suppose a particle moves in such a way that it covers different distances in equal time intervals. Its speed is then said to be variable and its speed at a particular instant is called the instantaneous speed. The speedometer in an automobile indicates the instantaneous speed of the automobile.
Likewise, a particle may have variable velocity. Velocity can vary by a change in magnitude, by a change in direction, or both. If it is so, then we must seek to determine the velocity of the particle at any given instant of time, called the instantaneous velocity. The instantaneous velocity can be obtained from the average velocity in the following way. We have
<v⃗ >=Δr⃗ Δt
If we go on decreasing the time interval Δt, then from this formula we shall be knowing the average velocity of the particle for smaller and smaller time intervals. When Δt becomes infinitesimally small (Δt→0) then from the above formula, we shall be knowing the velocity of the particle at a particular instant of time. Thus
Instantaneous velocity v⃗ =LtΔt→0Δr⃗ Δt=dr⃗ dt
Note that if we let Δt→0, the displacement vector approaches a limiting direction namely that of the tangent to the path of the particle at that instant. Thus the direction of the instantaneous velocity vector is always tangential to the trajectory of the particle The instantaneous speed can be defined as the magnitude of the (instantaneous) velocity vector.

Average Acceleration and Instantaneous Acceleration

Very often, the velocity of a particle changes either in magnitude, in direction, or both as the motion proceeds. The body is then said to have an acceleration. The acceleration is defined as the time rate of change of velocity. The average acceleration is defined as
<a⃗ >=v⃗ 2−v⃗ 1t1−t1=Δv⃗ Δt
The direction of average acceleration (vector) is the direction of change in velocity (vector). Its unit is m/s²
The instantaneous acceleration is defined by
a⃗ =LtΔt→0Δv⃗ Δt=dv⃗ dt
In general, there is no relation between the direction of velocity vector and the direction of acceleration vector.

Application 3

The position of a particle is given by x = 40 – 5t – 5t², where x is in metre and t is in second.
(a) Find the average velocity between 1 and 2s
(b) Find its instantaneous velocity at 2s
(c) Find its average acceleration between 1 and 2s
(d) Find its instantaneous acceleration at 2s

Solution

(a) At t=1s; xi=30m
t=2s; xf=10m
vav=xf−xitf−ti=10−302−1=−20m/s
(b) v=dxdt=−5−10t
at t=2s; v = -5-10(2) = –25 m/s
(c) At t=1s; v = -5-10(1) = -15 m/s
At t=2s; v = -5-10(1) = -25 m/s
vav=xf−xitf−ti=−25−(−15)2−1=−10m/s
(d) a=dvdt=−10m/s2