**Introduction**

**Motion:-**If an object changes its position with respect to a reference point with elapse of time, the object is said to be in motion.

**Rest:-**When an object does not change its position with respect to a reference point with elapse of time, the object is said to be in rest.

**Example –**When a vehicle changes its position with respect to an electric pole (a reference point) with elapse of time, then vehicle is called in motion. And if the same vehicle does not change its position with respect to that electric pole, the vehicle is called in the state of rest.

Thus to observe the motion of any object; two things are necessary – a reference point and time.

In the above example; vehicle is the object and electric pole is the reference point. A building, tree, or any other static thing can be taken as reference point to observe the motion of an object.

**Motion along a straight line:**

When an object moves along a straight line, the motion of the object is called rectilinear motion. For example; motion of a vehicle along a straight road.

**Distance and Displacement:**

Distance is the length of path covered by a moving object in the given time irrespective of direction. Distance has only magnitude and no direction.

__SI unit of distance is meter (m).__

Kilometer is used to measure bigger distance and it is abbreviated as ‘km’.

Displacement is the shortest possible distance covered by a moving object from initial point in a particular direction. In other words, shortest distance between initial point and final point is called the displacement.

Displacement has both magnitude and direction while distance has only magnitude.

Suppose, a ball is rolling along a straight line.

**Case – 1**

Suppose, the ball starts moving from point A and reaches at point B.

Thus, distance covered by ball = 10 m

Displacement of ball = 10 m towards west.

**Case - 2**

Suppose, ball starts moving from point A and reaches to B. Again it returns on the same path from point B and reaches at A.

Thus, distance covered by the ball = distance from A to B + Distance from B to A

= 10 m + 10 m = 20 m

In this condition, distance covered by ball = 20 m.

Since, ball returns at point A, thus displacement of the ball = 0

**Case – 3 –**

Suppose, the ball starts moving from point A, reaches point B and returns back to point C.

Then, the distance covered by ball = distance from A to B + Distance from B to C

= 10 m + 7 m = 17 m

Displacement of ball = Distance of point C from A = 3 m towards west.

**Uniform and Non-Uniform Motion:**

**Uniform Motion: –**When an object covers equal distance in equal interval of time, the motion is called uniform motion. For example – if a moving vehicle covers a distance of 10 km every hour, the motion of the vehicle is called uniform motion.

**Non-Uniform Motion: -**When an object covers unequal distance in equal interval of time, the motion is called non-uniform motion. For example – If moving vehicle covers a distance of 10 km in the first hour, covers a distance of 20 km in the second hour, covers a distance of 5 km in the third hour, etc. the motion of the vehicle is called non-uniform motion.

**Speed:**Distance covered by a moving object in unit time is called distance.

Where, v = speed, s = distance, t = total time.

__SI unit of speed is meter per second (m/s).__

**Average speed:**

sThe average distance covered in unit time by a moving object is called average speed. Average speed is the ratio of total distance covered and total time taken.

Where, v = Average Speed, s = Total distance covered, t = total time taken.

__SI unit of average speed is meter per second (m/s).__

**Velocity:**

The speed of a moving object in particular direction is called velocity. Velocity has both magnitude and direction while speed has only magnitude and no direction.

Velocity of an object is the distance covered in particular direction in unit time.

__SI unit of velocity is meter per second.__

**Uniform velocity:**

Uniform speed of an object in same direction is called uniform velocity.

Non-Uniform velocity: Velocity of an object is changed in following two conditions.

*(a) When speed is changed*

*(b) When direction is changed.*

Thus, non uniform speed of a moving object in same direction, or non-uniform speed in different directions or uniform speed in different directions is called non-uniform velocity.

**Example –**If a vehicle is moving on a circular path with uniform speed, then its velocity is said to be non-uniform, because on a circular path the direction of moving body changes along with direction of curve.

If a vehicle moving with uniform speed on a jig-jag path, the velocity of the vehicle will be non-uniform because direction of vehicle is changed with the change of direction of path.

**Average Velocity:**The arithmetic mean of velocity of an object moving along a straight line is called the average velocity.

Where, u is the initial velocity and v is the final velocity.

The displacement of a moving object in unit time is also called the average velocity.

**Acceleration:**The rate of change in velocity is called acceleration. Acceleration is generally denoted by ‘a’ or f.

Where, ‘a’ is acceleration, ‘v’ is final velocity, ‘u’ is initial velocity and ‘t’ is time taken for change.

A positive sign of the magnitude of acceleration shows increase in velocity and a negative sign show decrease in velocity. If there is decrease in acceleration, it is called Retardation. This means, rate of decrease in velocity is called Retardation.

__SI unit of acceleration:__

The SI unit of velocity is meter /second

The SI unit of time is second.

**Acceleration in the case of Uniform Velocity:**

**In the case of uniform velocity, the speed or direction of a moving object is not changed and thus there is no change in acceleration. Therefore, in the case of uniform velocity acceleration will be zero.**

**Equation of Motion:**

Relation among velocity, distance, time and acceleration is called equations of motion. There are three equations of motion:

**First Equation of Motion:**

The final velocity (v) of a moving object with uniform acceleration (a) after time, t.

Let

The initial velocity = u.

Final velocity = v.

Time = t

Acceleration = a

This equation is known as first equation of motion.

**Second Equation of Motion:**

Distance covered in time (t) by a moving body.

Let,

Initial velocity of the object = u

Final velocity of the object = v

Acceleration = a

Time = t

Distance covered in given time = s

We know that, Distance covered (s) in given time = Average velocity x Time

Or, s = Average velocity x Time -----------------(iii)

After substituting the value of average velocity from equation (ii) we get

After substituting the value of ‘v’ from first equation of motion we get,

The above equation is known as Second equation of motion.

**Third Equation of Motion:**

The third equation of motion is derived by substituting the value of time (t) from first equation of motion.

We know from first equation of motion,v=u+at

By substituting the value of ^' t^' from euqation (v) we get

This is called the Third equation of motion.

__Graphical Representation of Motion:__

**Distance – Time Graph:**

When an object is moving with uniform velocity, the slope of graph is always a straight line. In other words slope of straight line of a distance-time graph shows that object is moving with uniform velocity.

In the above graph, straight slope line shows that object is moving with uniform velocity. Slope OB shows the velocity of the object.

**Calculation of Velocity using distance-time graph:**

To calculate the velocity, let take two points A and B on the slope OB.

Draw one line parallel to y-axis and another parallel to x-axis from B.

Again draw a line parallel to y-axis and another parallel to x-axis from point A.

Let, line parallel to x-axis from point B cut at a point, S2 at y-axis.

Line parallel to x-axis from point A cut at point, S1 at y-axis.

Let, line parallel to y-axis from point B cut at t2 at x-axis.

Line parallel to y-axis from point A cut at t1 at x-axis.

Now, BC= Distance = S2 – S1 and AC = time = t2 – t1

We know that slope of the graph is given by the ratio of change in y-axis and change in X-axis.

__Distance – Time Graph of a body moving with Accelerated motion:__

When graph of distance Vs time is plotted for an object moving with accelerated motion, i.e. with increasing non-uniform speed, the slope of graph will not be a straight line. The rising trend of slope shows the increasing trend of velocity.

**Velocity-Time Graph:**

**Velocity :–**time graph of an object moving with uniform velocity:

The slope of a Velocity – time graph of an object moving in rectilinear motion with uniform velocity is straight line and parallel to x-axis when velocity is taken along y-axis and time is taken along x-axis.

__Calculation of distance using velocity-time graph:__

Let two points A and B on the slope of graph.

Draw two lines parallel to y-axis AC from point A, and BD from point B.

Let point D at the x-axis (time axis) is t2 and point C is t1.

Let AB meet at ‘v’ at y-axis, i.e. object is moving with a velocity, v.

Thus, distance or displacement by the object is equal to the area of the rectangle (shaded) ABCD.

Thus, Area of ABCD = BD x DC

⇒ s = v (t2 – t1)

Since given object is moving with constant velocity along a straight line, thus displacement will be equal to distance covered.

Therefore, Distance or Displacement = velocity X time interval.

__Velocity – Time Graph of an object moving with uniform acceleration:__

When velocity – time graph is plotted for an object moving with uniform acceleration, the slope of the graph is a straight line.

The pattern of slope of the graph shows that object is moving with uniform acceleration.

Calculation of Displacement and Distance covered by the moving object using velocity time graph:

Let take two points, A and B at the slope of the graph.

Draw a line from B to BD and another from point A to AE parallel to y-axis.

Let AD meets at t2 and AE at t1 on the time axis.

Thus, Distance covered by the object in the given time interval (t2 – t1) is given by the area of ABCDE.

Velocity time graph of an object moving with uniform decreasing velocity:

The slope of the velocity time graph of an object moving with uniform decreasing velocity with uniform acceleration is a downwards straight line. The straight downward slope shows the decreasing velocity with uniform acceleration, i.e. retardation.

Velocity time graph of an object moving with non-uniform velocity:

Jig – zag line of slope of graph shows that the object is moving with non-uniform velocity.

__Equation for Velocity – Time relation by graphical method – First equation of Motion –__

Let an object is moving with uniform acceleration.

Let the initial velocity of the object = u

Let the object is moving with uniform acceleration, a.

Let object reaches at point B after time, t and its final velocity becomes, v

Draw a line parallel to x-axis DA from point, D from where object starts moving.

Draw another line BA from point B parallel to y-axis which meets at E at y-axis.

Let OE = time, t

Now, from the graph,

BE = AB + AE

⇒ v = DC + OD (Since, AB = DC and AE = OD)

⇒ v = DC + u (Since, OD = u)

⇒ v = DC + u ------------------- (i)

Above equation is the relation among initial velocity (u), final velocity (u), acceleration (a) and time (t). It is called first equation of motion.

__Equation for distance –time relation:__

Distance covered by the object in the given time ‘t’ is given by the area of the trapezium ABDOE

Let in the given time, t the distance covered by the moving object = s

The above expression gives the distance covered by the object moving with uniform acceleration. This expression is known as second equation of motion.

Equation for Distance Velocity Relation: Third equation of Motion:

The distance covered by the object moving with uniform acceleration is given by the area of trapezium ABDO

Therefore,

The above expression gives the relation between position and velocity and is called the third equation of motion.

While solving the problems related to velocity, distance, time and acceleration following three points should be considered:

**Motion along a circular path:**

Motion of an object along a circular path is called circular motion. Since, on a circular path the direction of the object is changing continuously to keep it on the path, the motion of the object is called accelerated motion.

Velocity in the case of circular motion.

If the radius of circle is ‘r’

Therefore, circumference = 2πr

Let time ‘t’ is taken to complete one rotation over a circular path by any object

Where, v = velocity, r = radius of circular path and t = time

Motion of earth around the sun, motion of moon around the earth, motion of a top, motion of blades of an electric fan, etc. are the examples of circular motion.

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