Chapter – 3
Motion and Straight Line
Mechanics: The branch of physics which deals with the study of the motion of the object and the equilibrium of the object under many forces is known as mechanics. It consists of three branches. (i) Statius (ii) Kinematics (iii) Dinamics
 Status: The branch of mechanics which deals with study of the objects or in equilibrium under the action of external force.
 Kinematics: The branch of mechanics which deals with the study of the mechanical motion of the object without taking into the account of the cause of the motion in the object or body.
 Dinamics: The branch of mechanics which deals with the study of the mechanical motion of the bodies by taking into the account of the cause of the motion in the body.
Frame of reference: The fixed point or place with respect to which the position, velocity, acceleration, etc of a body or an object is measured is known as frame of reference.
Types of Frame of reference: The various types are as follows:
 Cartesian coordinate system.
 Spherical, polar coordinate system.
 Cylindrical coordinate system.
A Cartesian coordinate system is of two types:
 Innertial frame of reference.
 NonInnertial frame of reference.
 Innertial frame of reference: Any frame of reference which is either at rest or moving with velocity is called Innertial frame of reference. In Innertial frame of reference Newton’s first Law of motion is valid or true.
 Accelerated or non Innertial frame of reference: The frame of reference which is accelerated is called nonInnertial frame of reference. In nonInnertial frame of reference Newton’s first Law of motion is not valid.
Types of Motion
 Translational Motion: A body has translational motion if a line joining any two points remains parallel to itself throughout the motion of the body. E.g. (i) A bus moving on a straight road, (ii) A ball falling vertically downwards from a certain height.
Notes: (i) during translation motion of the body, there is change in location of the body; (ii) It is also non as rectilinear motion.
 Rotational Motion: A body has rotational motion if it moves around at fixed line from axis of rotation. Such that all the reticules constituting the body move simultaneously along concentric circle having their centres on the axis of rotation. E.g. Motion of earth around the sum.
Note: During rotational motion of a body there is change in the orientation of the body while there is no change in the location of the body from the axis of rotation.
 Vibrational motion: A body has vibrational motion if it moves to and fro after regular interval of time about a fixed point called the mean position us the equilibrium position. E.g. Motion of the pendulum of a wall clock.
 One dimensional motion: A particle moving along a straight line or a path is said to have one dimensional motion.
 Two dimensional motions: A particle moving in a plane is said to have two dimensional motions.
 Three dimensional motions: A particle moving in spouse is said to have three dimensional motions.
Distance and displacement
The length of the actual path travel by a particle between initial and final position of a particle in a given interval of time is called distance covered by the particle. The S.I. unit is metre (m).
 It is a scalar physical quantity as it has only magnitude and no sense of direction.
 Distance travel by a particle in a given time interval is always positive.
 It can neither be negative or 0.
 Distance travel by a vehicle is measured by a device non as odometer
Displacement
The shortest distance from the initial position to the final position of the particle is called displacement. The S.I. unit is metre.
 Displacement is a vector physical quantity as it has both magnitude and direction.
 Displacements can – ve, + ve or 0.
Difference between distance and displacement
Distance

Displacement

1. It is the length of the actual path travel by the particle.
2. It is always positive.
3. Distance is a scalar physical quantity.
4. Distance travel by a particle depends upon the shape of the path followed by the particle.

1. It is the shortest length between the initial and final position.
2. It may be + ve, or – ve, or 0.
3. Displacement is a vector physical quantity.
4. Displacement of a particle does not depend shape of the path followed by the particle.

Average speed and Average velocity
Speed: the distance travel by a body per unit time is known as speed. i.e.
The S.I. unit of speed is
Q. Concert 1 KM/hr into M/S.
Ans.
Uniform speed: A particle is said to be moving with uniform speed, if it causes equal distance or equal interval of time.
Variable speed: A particle or a body is said to be moving with variable speed or nonuniform speed, if it covers unequal distance in equal interval of time.
Average speed: Average speed of a body is defined as the ratio of the total distance travel by its to the total time taken.
Let,
Example No. 3.2: The position of an object moving along xaxis is given by, whereand t is measured in second what is its velocityand. What is the average velocity betweenand.
Average speed in different situation:
 Particle covering different distance with different speed.
 If a body covers equal distance with speedand respectively, then average speed of the body is given by
 Body moving with different speed in different time interval.
Let a body move with velocitiesand so on at the time intervalsand so on.
Total distance covered
Hence average speed of the body,
If
Q. A car covers first half of the total distance with a speed of 36 Km/h and a second half with a speed of 54 Km/h. Find the average speed of the car.
Velocity: Distance of a particle or a body per unit time is called the velocity of the body.
i.e.
Uniform velocity: A body is said to move with uniform velocity if it covers equal displacement in equal interval of time.
Non uniform velocity: A body is said to move with variable velocity. If it covers equal displacement in unequal interval of time
Average velocity:
Q. A boy runs from his home to the market 1 Km away. He reaches the market in 30 minutes. Seeing the market closed, he at ones runs back to his home and stops after 15 minutes. At his friends house which is half Km away from the market. Calculate his average speed and magnitude of average velocity.
Average velocity,
Instantaneous Velocity and Speed
Instantaneous speed: When a body is moving with variable speed then the speed of the body at any instant of time is called instantaneous speed.
Hence the instantaneous speed of a body is defined as the time rate of change of distance travelled by it.
Instantaneous velocity: It is defined as the velocity of the particle of any instant or at any point of its path of motion.
Instantaneous velocity is equal to limiting values of average velocity as
Whereandare infinite valuesandrespectively. Hence, instantaneous velocity of a particle known as the time rate of change of the displacement of the particle.
Q. A body is moving in a straight line along xaxis. Its distance from xaxis is given by equation, whereisand it is. Find average speed of the cloudy on time interval to and = 20 and its instantaneous speed at .
Difference between velocity and speed
Speed

Velocity

1. It is the distance travelled by a body per unit time.
2. Speed of a body tells nothing about the direction of motion of the body.
3. Speed of a body is always + ve.
4. Speed of a body is equal to or greater then the velocity of the body.
5. Speed is sealer physical quantity.

1. It is the displacement of the body per unit time.
2. Velocity tells the direction of motion of the body.
3. Velocity of a body may be + ve or – ve or zero.
4. Velocity of a body is equal or less then the speed of the body.
5. Velocity is a vector physical quantity.

Determine of velocity from displacement time graph.
Consider two points A and B m the displacement time graph. Letbe the time andbe the displacement corresponding to points A. Letbe the time andbe the displacement corresponding to the point B. Now the displacement of the body in time intervalequal to.
Acceleration: The rate of change of velocity of an object with time is called the acceleration of the object. i.e.
Uniform acceleration: The motion of an body or an object whose acceleration is constant is known as uniformly accelerated motion.
Won uniform acceleration: The motion of an object having variable acceleration is known as non uniform acceleration motion.
Q. Draw velocity time graph when accelerated = 0.
Physical meaning of acceleration: The physical meaning of acceleration is that it tell us by how much amount of the velocity of an object is changed per unit time.
Average acceleration: It is defined as the changed in velocity of the object divided by the total time taken for this change in velocity i.e. average acceleration.
Itbe the change in velocity in timethen
Instantaneous acceleration: It is defined as the accelerated of an object at any instant of time during its motion. i.e.
Thus acceleration of an object is equal to the time rate of change of velocity of the object.
Q. A particle in moving straight line displacement at any instantis given byfind the average acceleration on the time intervalandand instantaneous acceleration at
(ii)
Velocity time graph for uniformly accelerated motion: Determine the acceleration using velocity time graph.
Velocity time relation is given by
Whereis initial velocity andis also a constant
Determination of acceleration of the particle consider two points A and B on the velocity time graph as shown in the figure above.
Letbe the velocity of the particle of instantto prosponding to point A. Letbe the velocity of the particle at instant prosponding to point B.
Change in velocity, when particles moves from point A to point B
Time taken for this change in velocity
Acceleration,
Thus acceleration of a uniformly accelerated motion of the particle is equal to the slope ofgraph.
Q. An expression of physical quantity is written as, whereis the applied force andis the mass per unit length. Find the dimensional representation ofand identify the physical quantity.
Ans.
Q. The motion of a particle of massis given byfind the force acting on the particle.
Ans.
Kinematic equation or relation for uniformly accelerated motion (graphical method): When a particle move in a straight line with constant acceleration then the portion, time, velocity and acceleration of the particle are represented by equation known as Kinematic equation of motion.
(i) Velocity attained after time: Letbe the initial velocity of the particle atandis the final velocity of the particle after time. Consider true points A and B on the curve corresponding toandrespectively. Draw BDon time axis. Also draw ACon BD.
In vector form equation can be written as Displacement
(2) Travel in time interval f:
Let= position of the particle atfrom the origin.
=position of the particle atfrom the origin.
=distance travelled by the particle in time interval
We know distance travelled by a particle in the given time interval = area under velocity time graph.
Divide rule of differentiation:
Q.
Integration: Integration is a inverse pro of differentiation
(i)
(ii)
Q.
Q.
Q.
In vector form equation (B) can be written as
(3) Velocity attained after travelling a distance S: We know distance travel by a particle on time t is equal to the area under velocity time graph.
The distance S travelled by a particle during time interval t is given by
Kinematic equation of motion for uniformly accelerates motion (Calculus method):
(i) Velocity attained by a particle after time t.
Letbe the change in velocity of a particle in time. Therefore, acceleration of the particle is given by
(ii) Displacement of the particle after time t:
Letbe the displacement of the particle in time
(iii) Velocity attain by a particle after travelling a distance S.
(iv) Displacement of the particle insecond:
Letandbe the displacement of the particle. Inandseconds respectively. Then the displacement of the particle insecond is given by
We know,
Q. A particles moves 10 m in 4th second and 15 m in 6th second of its motion. Calculate the distance travel by it in 7th second if the motion of the particle is uniformly accelerated motion.
Ans. We Know
Given
When,
Freely falling bodies: The motion of a body falling under the effect of gravity alone with a constant acceleration is known as free fall.
Equation for accelerated motion applies to freely falling body. Therefore, Equation of motion of the freely falling bodies can be obtained by thing g in place of ‘a’ in equations for accelerated motion a thus, for a freely falling body.
Special Case: If a body released or dropped from certain height, then initial velocity of the body is taken a zero, i.e. u = O.
Stepping distance of Vehicles: We know. When moving vehicles stops, becomes
Equation (ii) gives the stopping distance. It is clear from equation (ii) that stopping distance is directly proportional to square of the initial velocity if negative acceleration of vehicle is constant. i.e. stopping distance directly.
Relative velocity: Relative velocity of an object A w.r.t another object B is the time rate in which the object A changes its position with respect to object B.
Let, = displacement of A at.
= displacement of B at.
Now, letandbe the displacement of A and B respectively after time t.
So the displacement of the object A in time t is given by
For B in time t
Subtracting equation (i) from equation (ii) we get
Now, the relative displacement of B with respect to A at
the relative displacement of the object B with respect to object A at instant t
equation (iii) →
in which is the relative velocity of objects B w.r.t to the object it.
Special Case: (i) When the two object move with equal velocity i.e. equation.
(ii) When the two objects move with unequal velocities (a) when>then from equationis. This shows the separation between two moving object will go on decreasing with time.
(b) When >then from equation (4) is. This shows the separation between two object will go on increasing with time.
(iii) When two object moves is opposite direction, Let object A moves alongdirection with velocityand object B moves alongdirection with velocity.
Therefore, in this caseis taken as positive andis taken as. Hence relative velocity of object A w.r.t object B is given by
Extra:
 Newtonian mechanics or classical mechanics: It deals with the study of motion of the object moving with speeds much less then the speed of light in air.
 Relativistic mechanics: It deals with study of motion of the object moving with speed comparable to the speed of light in air.
Graphs: Velocity time graph for uniform positive acceleration:
For uniformacceleration:
Graph for motion for body having uniformacceleration:
For uniformacceleration: