Distance, Displacement, Speed and Velocity

Introduction 
        Kinematics is the science of describing the motion of objects using words, diagrams,numbers, graphs, and equations. Kinematics is a branch of mechanics. The goal of any study of kinematics is to develop sophisticated mental models  that serve to describe (an ultimately, explain) the motion of real-world objects.In this lesson, we will investigate the words used to describe the motion of objects.Thatis, we will focus on the language of kinematics. The hope is to gain a comfortable foundationwith the language that is used throughout the study of mechanics. We will study such termsas scalars, vectors, distance, displacement, speed, velocity and acceleration. These words are used with regularity to describe the motion of objects. Your goal should be to become very familiar with their meaning.
Mechanics : The branch of physics which deals with the study of force and motion their relatationship is called mechanics.The study of mechanics is divided into three parts.
     i) Statics: the branch of mechanics which deals with objects at rest is called statics. 
     ii) Kinematics : Kinematics which is derived from a Greek word kinema meaning motion, is a branch of physics,the branch of mechanics which deals with the motion of objects onlywithout considering  the cause of motion is called kinematics.
    iii) dynamics: the branch of mechanics which deals with the cause of motion is called dynamics.
Rest: A body is said to be at rest if it does not change its position with respect to thereference point. The objects which remain stationary at a place and do not change theirposition are said to be at rest.The position of a body with respect to surroundings does not change with time, the body is said to be in the state of rest.
    Ex: A chair lying in a room is in the state of rest, because it doesn’t change its positionwith respect to the surroundings of the room.A tree, An electric pole, our house, our school....... etc.
Motion: A body is said to be in motion if it changes its position with respect to the 
surroundings with the passage of time. All moving things are said to be in motion.All movingthings are said to be in motion. 
    Ex: A moving car, a moving train, a flying bird ...etc.
Rest and motion are relative terms : Rest and motion are relative terms. A body can be at rest as well as in motion at the same time. 
For example, when a bus moves on a road, then the bus as well as the passengers sitting in it change their position with respect to a person standing on the road side. So the bus and the passengers sitting in it are in motion with respect to the person standing on theroad side. However, the passengers sitting in the bus do not change their positions withrespect to each other. It means, the passengers sitting in a moving bus are not in motion withrespect to each other. Ex : A person sitting in the compartment of a moving train is in the state of rest, withrespect to the surroundings of compartment. Yet he is in the state of motion, if he compareshimself with surroundings outside the compartment
Scalars:The physical quantities which have only magnitude but not direction are called scalars.
   Ex: Mass, length, distance, time, area, volume, density, work etc
Vectors: The physical quantities which have both magnitude and direction are called vectors.
   Ex: Displacement, velocity, acceleration, force  etc
Distance:   The path length covered by the body is called distance.it is a scalar quantity.
    Units: cm (In C.G.S. System);  m(In S.I. System
Displacement: The shortest pathlength between the initial and final positions of a body iscalled displacement. It is a vector quantity.
            Units: cm (In C.G.S. System);  m(In S.I. System)
Examples:
1)  Suppose a bus starting from station A travels 15000 m to reach stationB then the distance covered by the bus is 15000 m. Now if the bus returns to the station A  then distance covered is 15000 m and the total distance covered by the bus during the trip from A to B and then back to A from B is 15000 m + 15000 m = 30000 m. 
                                                                      
                                                                   A bus moving from A to B and again from B to A
        But the displacement when the bus moves from A to B and then from B to A is zero
2) Suppose a person moves 3 meters from A to B and 4 meters from B to C as shown in the figure. The total distance traveled by him is 7 meters and he is displaced only by 5 m which is the shortest distance between his initial position and final position.
                                                                        
3)Now let us consider an object changing its position, with respect to a fixed point called the origin 0. x_i and x_f are the initial position and final position of the object. Then the displacement of the object = x_f – x_i. 
Case 1
    Suppose the object is moving from +1 to +4, then displacement =x_f – x_i = +4 – (+1)  = +3
                                                                   

Case 2
    If the object is moving from -3 to -1 then displacement =x_f – x_i  = –1 – (–3)= 2
                                                                 
Case 3
    If the object is moving from +5 to +2 then displacement =x_f – x_i = +2 – (+5) = –3.
                                                            
Case 4
    If the object follows the path as shown in the figure then the final position and the initialposition is the same i.e., the displacement is zero.  
                                                                     
    
Getting Direction
On the Earth the directions parallel to the ground which are North, East,West & south directions are horizontal and they are represented on paper as in side figure. The direction exactlly midway between N and E is called NE. Similarly NW, SW & SE. If the directions don't fall exactly midway then they won't be represented as NE, NW, SE, SW & SE. They are represented as in the  following example.
                                                                                       
                                                                                   
WORKED EXAMPLES
Example-1:An athlete completes one round of a circular track of radius R in 40 s. What will be his displacement at the end of 2 min 20 s? 
Sol.    The time = 2 min 20s = 140s

In 40 seconds athlete completes = 1 round
        In 140 seconds  athlete will completes 
          = \(140 \over 40\)round =3.5 rounds    
        The displacement in 3 rounds = 0
So net displacement =2R
Speed: The distance travelled by the body in unit time is called its speed.
        Speed (V) = \(\frac{Dis\tan ce\,\,travelled}{Time\,\,taken}\)
         *  speed is a scalar quantity.         
         * it is represented by v or u
    units: CGS Unit : cm/s,     SI unit: m/s
Uniform speed: If a body travels equal distances in equal intervals of time then it is said tobe moving with uniform speed.
    Eg: motion of ball on a frictionless plane surface.
Non - uniform speed: If a body travels unequal distances in equal intervals of time(or)equal distance in unequal intervals of time the body is said to be travelling with nonuniform (or) variable speed.
Instantaneous speed : The speed of a body at any instant known as the instantaneousspeed.speedometer of vehicle measures the instantaneous speed. 
Velocity: The rate of displacement (or) displacement per unit time is called velocity.        
    * The velocity of a body can never be greater than the speed of that body.
Velocity \(\left( \overset{\to }{\mathop{v}}\, \right)\,=\,\frac{Displacement}{time}=\frac{\overrightarrow{S}}{t}\)
    * velocity is a vector quantity.
    units:CGS Unit : cm/s, SI unit: m/s, 
    Note:* The velocity of a body can be zero, negative or positive.
    * The numerical value of velocity of a body can be equal to speed only if the body is moving along a straight line in the same direction.
Uniform velocity: If a body travels equal displacements in equal intervals of time then thebody is said to be travelling with uniform velocity.
Non - uniform (or) variable velocity: If a body covers cover unequal displacements in equal intervals of time then it is said to be travelling with variable velocity.