# Physics 9th: Motion | Graphical Representation of Motion

## Physics 9th: Motion | Graphical Representation of Motion

### Graphical Representation of Motion

A Graph represents the relation between two variable quantities in pictorial form. A graph is plotted between two variable quantities.

The quantity that is made to alter at will is called independent variable. The other quantity, which varies as a result of this change is called the dependent variable.

These graphs are used to calculate speed, acceleration and distance.

For the purpose of graphical representation speed and velocity are taken in the same meaning.

There are two types of graphs we draw

1.    Distance –Time graph

2.    Speed – Time graph or Velocity-Time graph

### DISTANCE –TIME GRAPH

(a) When we draw Distance –Time Graphs, time is always taken on the x-axis and distance on the y-axis.

In this graph AB is a straight line parallel to x-axis. In this graph, the distance travelled is not increasing with time. This is a graph of stationary object.

(b) The Distance-Time graph of a body moving at uniform speed is represented by a straight line OA as shown in figure. It can be used to calculate the speed of the body by finding the slope of the graph

$\displaystyle Speed=\frac{Dis\tan ce\,\,travelled}{Time\,\,taken}$

$\displaystyle Speed=\frac{AB}{OB}$

$\displaystyle \frac{AB}{OB}$ is also taken as the slope of the graph, which indicates the speed of the body.

(c) When the Speed – Time graph is a curve, it represents non-uniform motion. This curved line is called a PARABOLA

### SPEED-TIME GRAPH

#### When speed remains constant (zero Acceleration)

If the speed-time graph of a body is a straight line parallel to the time axis, then speed of the body is constant (uniform)

$\displaystyle \text{Speed}=\frac{\text{Distance}}{\text{Time}}$

$\displaystyle \text{Distance}=\text{Speed}\times \text{Time}$

= YZ ´ OZ

Therefore, the area enclosed by the speed-time graph and the x-axis(time-axis), represents the Distance travelled by the body.

#### When speed increases at uniform rate (uniform Acceleration)

The speed –time graph for uniformly changing speed will be a straight line.

Acceleration $\displaystyle =\frac{change\,\,in\,\,speed}{Time\,\,Taken}$

Acceleration $\displaystyle =\frac{BC}{OC}$

Therefore acceleration of a body is given by the slope of the graph $\displaystyle \left( i.e.\frac{BC}{OC} \right)$

The distance travelled = Area under DOBC = $\displaystyle \frac{1}{2}\times OC\times BC$

#### When speed decreases at uniform rate(Retardation)

In speed time graphs, a straight line sloping downwards indicates uniform retardation.

#### When initial speed of the body is not zero

When the initial speed of the body is not zero, the speed-time graph is represented as a straight line forming a trapezium with the time-axis.

Distance travelled     = Area of Trapezium OXYZ

= $\displaystyle \frac{1}{2}$ (sum of parallel sides) (Height)

$\displaystyle =\frac{1}{2}\left( OX+YZ \right)\left( OZ \right)$

$\displaystyle =\frac{1}{2}\left( u+v \right)\left( t \right)$

#### When speed changes at non-uniform rate

Speed-time graph of non-uniform speed (non-uniform acceleration) is a curved line.

# Physics 9th: Motion | Acceleration and Retardation

## Physics 9th: Motion | Acceleration and Retardation

### ACCELERATION

We notice that bodies do change their speed or velocity. For example if we see the speedometer of a car, we will notice that the needle keeps on moving, that is the velocity of the car keeps on changing. The rate at which this change in velocity occurs, is known as acceleration.

\     Acceleration of a body can be defined as the rate of change of velocity with time

$\displaystyle Acceleration=\frac{Change\,\,in\,\,velocity}{Time\,\,taken\,\,for\,\,change}$

$\displaystyle Acceleration=\frac{Final\,\,velocity-Initial\,\,velocity}{Time\,\,taken}$

$\displaystyle a=\frac{v-u}{t}$

Where         a = Acceleration of the body

v = Final velocity of the body

u = Initial velocity of the body

t = Time taken for this change in velocity

·    S.I. unit of acceleration is $\displaystyle m{{s}^{-2}}$ (meter per Second Square)

·    Acceleration is a vector quantity

·    When a body is moving with uniform velocity, its acceleration is zero (or no acceleration)

This is because there is no change in velocity as initial velocity = Final velocity.

·    A body moving with non-uniform velocity is said to be in accelerated motion.

### Question:     Calculate the acceleration of a body which covers a distance of 50 m in every second while moving on a straight road.

Solution:

Velocity (A to B)$\displaystyle =\frac{50}{1}$

(u) = 50 m/s

Velocity (B to C)$\displaystyle =\frac{50}{1}$

(v) = 50 m/s

$\displaystyle a=\frac{v-u}{t}=\frac{50-50}{2}=0$

Therefore, in case of uniform motion, acceleration is zero

### Question:     Calculate the acceleration of a body which covers a distance of 50 m in one second and 70 m in next one second while moving on a straight road.

Solution:

$\displaystyle velocity=\frac{50}{1}$

(u) = 50 m/s

$\displaystyle velocity=\frac{70}{1}$

(v) = 70 m/s

$\displaystyle a=\frac{v-u}{t}=\frac{70-50}{2}=\frac{20}{2}=10$ m/s2

#### Uniform acceleration

A body has uniform acceleration when it travels in a straight line and its velocity increases or decreases by equal amounts in equal intervals of time.

It can also be said that when the velocity of a body changes at a uniform rate, it is said to have uniform acceleration.

#### Non-uniform acceleration

A body is said to have non-uniform acceleration if its velocity increases or decreases by unequal amount in equal intervals of time

OR

When the velocity of a body changes at unequal rate or non-uniform rate.

#### Negative acceleration (retardation or deceleration)

Till now we have seen the case of increasing velocities. In many cases, there is also a decrease in velocity. For example when we apply brakes, the car stops after some time.

Therefore, if the velocity of the body increases, Acceleration is said to be Positive Acceleration

And if the velocity of a body decreases, acceleration is said to be Negative Acceleration, which is also called as Retardation or Deceleration.

Retardation is measured in the same way as acceleration.

$\displaystyle Retardation=\frac{v-u}{t}$

·    S.I. unit of retardation is $\displaystyle m{{s}^{-2}}$

·    Value of retardation is always negative as in this case ‘u’ is always larger than ‘v’