Graphical Representation of Motion is one of the very important topic for CBSE class 9 students. It is the first lesson where students learn to convert theory with graph. Some students feel it very hard. But some one feel very interesting. These second category students feel interesting because they learn properly. So here I try to give step wise proper guidance.

Before learn graphical representation of motion, we need to learn some topic which is related to it. In graphical method we need clear idea about

- Graph
- Plotting a graph
- Uses of graph

#### Graph

Simple telling a graph is a line. It may be straight line or curved. And most importantly it showing the relation between two variable quantities of which one varies as a result of the change in the other. The quantity which changes independently is called independent variable. And the one which changes as a result of the change in the other is called dependent variable.

#### Plotting a graph

- Take independent variable along X-axis and dependent variable along Y-axis.
- Choose convenient scale so that more than 2/3rd of graph is filled.
- Draw free hand curve to join them.

#### Uses of graph

- It gives a bird’s eye view of the changes.
- It is used to show dependence of one quantity on the other e.g., distance or velocity on time.
- We can find distance covered in a given time.
- Slope of velocity-time graph gives acceleration.
- We can find position or velocity of body at any instant.

So now come to our main topic ” Graphical Representation of Motion CBSE Class 9 “. Here we will learn conventional one and some special cases also.

#### Distance/Displacement – Time graph

This graph is plotted between the time taken and the distance covered, the time is taken along the x-axis and the distance covered is taken along the y-axis.

Speed =

- The slope of the distance-time graph gives the speed of the body.
- The slope of the displacement-time graph gives the velocity of the body.

##### When the body is at rest

If the position of the body does not change with time then it is said to be stationary, the distance-time graph of such a body is a straight line parallel to x-axis.

##### When the body is in uniform speed

As we know when the position of the body changes by equal intervals of time then body is said to be moving with uniform speed. The distance-time graph of such a body is a straight line, inclined to x-axis as per graphical representation of motion.

Slope = measure on y-axis / measure on x-axis

Slope =

= velocity V = . O R = speed

**Special case-I**

In uniform motion along a straight line the position x of the body at any time t is related to the constant velocity as,

x_{A} = vt Starting form zero

x_{B} = x0 + vt starting from x_{0}

**Special case-II**

Slope of line A = tanθ_{A} = tan0 ( θ_{A} = 0)

= zero velocity

Slope of line B = tanθ_{B} = positive velocity

Slope of line C = tanθ_{c} = more positive velocity

θ_{C} > θ_{B} (tanθ_{C} > θ_{B} )

Then v_{C} > v_{B}

Slope of line D = tan (– θ_{D} ) = negative velocity.

**When the body is in motion with a non-uniform (variable) speed**

The position-time graph is not a straight line, but is a curve. The speed of the body at any point is known as instantaneous speed and can be calculated by finding the slope at that point.

So instantaneous speed of the body at point A.

Slope at point A = tanθ_{A} = AE / CE

instantaneous speed of the body at point B

Slope at point B = tanθ_{B} = BF / DF

θB > θA so slope at point B is greater than the slope at point A.

Hence speed of body at point B is a greater than, the speed of body at point A.

**When the speed decreases with passage of time –**

Slope at point A > slope at point B (∴ θ_{A }> θ_{B })

So, speed at point A > speed at point B

#### Important note

A distance time graph can never be parallel to y-axis (representing distance) because this line has slope of 90° and slope = tanθ = tan90° = infinite, which means infinite speed. It is impossible.

##### Acceleration from displacement-time graph

**Line A :–** A straight line displacement-time graph represents a uniform velocity and zero acceleration

**For line B :–** A curved displacement-time graph rising upward represents an increasing velocity and positive acceleration

**Line C :–** A curved displacement-time graph falling downwards, represents a decreasing velocity and negative acceleration.

#### VELOCITY-TIME GRAPH

The variation in velocity with time for an object moving in a straight line can be represented by a velocity-time graph. In this graph, time is represented along the x-axis and velocity is represented along the y-axis.

Acceleration =

Hence the slope of the speed/velocity-time graph, gives the acceleration of the body.

Distance = speed × time

Hence, area enclosed between the speed-time graph line and x-axis (time axis) gives the distance covered by the body. Similarly area enclosed between the velocity-time graph line and the x-axis (time axis) gives the displacement of the body.

**Note:** Since the graph takes into account, only the magnitude hence velocity-time graph is not different from speed-time graph.

**When the body is moving with constant velocity**

When the body moves with constant velocity i.e. its motion is uniform.

The speed or velocity of the body is uniform hence the magnitude remains same. The graph is a straight line parallel to x-axis (time-axis). Since the velocity is uniform. Its acceleration is zero. The slope of the graph in this case is zero.

**CONCLUSION :** Velocity-time graph of a body moving with constant velocity is a straight line parallel to time axis.

**When the body is moving with a uniform acceleration.**

The speed or velocity is changing by equal amounts in equal interval of time, the speed or velocity time graph of such a body is a straight line inclined to x-axis (time-axis).

**When the body is moving with a non-uniform (variable) acceleration**

As per Graphical Representation of Motion, the speed or velocity-time graph is not a straight line but is a curve.

The line has different slopes at different times, its acceleration is variable. At point A, slope is less hence acceleration is less. At point B slope is more hence acceleration is more.

**Note:** Speed or velocity-time graph line can never be parallel to y-axis (speed axis), because slope angle becomes 90° than tan90° is infinite it is impossible.

**Distance or displacement from speed or velocity-time graph**

As distance or displacement = speed or velocity x time, hence the distance or displacement can be calculated from speed or velocity-time graph.

**When speed or velocity is uniform (constant**)

Distance/displacement = Area of rectangle ABCD = AB × AD

Thus, We find that the area enclosed by velocity-time graph and the time axis gives the distance travelled by the body.

**When acceleration is uniform (constant)**

distance or displacement = area of right triangle OAB = ½ x base x height = ½ x OB x BA

**When speed or velocity as well as acceleration is non-uniform (variable).**

The speed-time graph of a body moving irregularly with variable speed and acceleration. For a small interval of time ∆t, as there is not much change in speed, hence the speed can be taken as constant.

∴ For this small time interval.

Distance ∆s = v∆t = Area of the blackened strip.

For whole time interval between t_{1} and t_{2} , distance = sum of areas of all the strips, between t1 and t2 = area of shaded figure ABCD.

**Application of Velocity- time Graphs : **

A number of useful results can be deduced from velocity time graph.

- Slope of velocity-time graph gives the acceleration.
- Area below velocity-time graph and the time axis gives the distance covered.
- Using the above two results, we can derive all equations of motion.

##### Let’s check how you learn Graphical Representation of Motion ?

`[WpProQuiz 6]`