Graphical Representation of Motion: The concept of the graphical representation of motion provides an easy way to understand and analyse the motion of an object. Representing the data in a graph provides a simpler method to get the basic information about the various physical quantities involved in an event. We use line graphs to represent the motion of an object.

In the graphical representation of motion, the dependent quantity is taken along the \(Y\)-axis and the independent quantity is taken on the \(X\)-axis. Check out this article to learn more about the graphical representation of motion Class 9, its types, uses, and more.

What is Graphical Representation of Motion?

Graphical representation of motion uses graphs to represent an object’s motion. It is a convenient way to describe the nature of an object’s motion. This approach to expressing an object’s motion makes it simple to examine the change in the many physical quantities of an object in motion. Graphs are extremely useful in examining a body’s linear motion.

For example, the slope of a displacement-time graph indicates the velocity (and speed) with which the object is moving. It also gives an idea of whether the motion of the object is uniform or non-uniform.

Graphical Representation of Motion – Definition

The graphical representation of motion is a method of representing a set of variables (physical quantities) pictorially with the help of a line graph where one physical quantity depends on the other physical quantity.

Types of Graphical Representation of Motion

The three main types of graphical representation of motion are displacement versus time graph, velocity versus time graph, and acceleration versus time graph.

A. Displacement-Time Graph

In a displacement-time graph, the displacement is plotted on the \(Y\)-axis, and the time is plotted on the \(X\)-axis. The displacement-time graph represents the change in the position of an object with respect to time.

1. Displacement-time Graph for a Body at Rest

An object is said to be at rest when it does not change its position with respect to time. The displacement-time graph for a stationary body will be a straight line parallel to the time axis. The figure shown below represents the displacement-time graph for a body at rest.

2. Displacement-time Graph for a Body Moving with Uniform Velocity

When an object covers equal distances in equal intervals of time and travels in a straight line, the object travels with a uniform velocity. The displacement-time graph of a body moving with uniform velocity will be a straight line inclined to some angle with the time axis, as shown in the figure below.

3. Displacement-time Graph for a Body Moving with Non-uniform Velocity

When an object covers unequal distances in equal intervals of time in a particular direction or when it is changing its direction of motion, then it is said to be moving with a non-uniform velocity. In this case, the displacement-time graph will be a curved line. The displacement-time graph of a body moving with non-uniform velocity is shown in the figure below.

The slope of the displacement-time graph gives the velocity of the object. To calculate the velocity of the object, we use the following formula:

\(v = \frac{{\Delta s}}{{\Delta t}} = \frac{{{s_2} – {s_1}}}{{{t_2} – {t_1}}}\)
Where \(v = \) velocity of the object, \(s = \) displacement covered by the object and \(t = \) time taken.

B. Velocity-Time Graph

The change in velocity of the body with time is represented by the velocity-time graph. In the velocity-time graph, the velocity of the body is represented along the \(y\)-axis, and the time is represented along the \(x\)-axis. The slope of the velocity-time graph gives the acceleration of the moving body.
\(a = \frac{{\Delta v}}{{\Delta t}}\)
Where, \(a = \) acceleration of the body, \(v = \) velocity of the body and \(t = \) time. The area enclosed under the velocity-time graph gives the displacement of the body.

1. Velocity-time Graph for a Body Moving with a Uniform Velocity

When a body moves with a uniform velocity, its magnitude and direction remain the same with time. The velocity-time graph for such a motion will be a straight line parallel to the time axis. In this case, the acceleration of the body is zero.

2. Velocity-time Graph for Bodies whose Velocities are Increasing and Decreasing with Time at a Constant Rate

The velocity-time graph for a body whose velocity is increasing at a uniform rate will be a straight line as shown in figure \(\left( f \right).\) The acceleration, in this case, will be positive. The velocity-time graph for a body whose velocity is decreasing at a uniform rate is shown in figure \(\left( g \right).\) The acceleration of the body will be negative if its velocity decreases with time.

3. Velocity-time Graph for a Body Moving with Variable Velocity

If a body is moving with a variable velocity, then the velocity-time graph for the body will be a curved line. In this case, the acceleration of the body is also changing with time.

C. Acceleration-Time Graph

The slope of the acceleration-time graph gives the quantity called the jerk. Jerk is the rate of change of acceleration.
\({\text{Jerk}} = \frac{{\Delta a}}{{\Delta t}}\)

The area under the acceleration-time graph represents the change in velocity.

Uses of Graphical Representation of Motion

The uses of graphical representation of motion are as follows:

1. The displacement-time graph indicates how the displacement of a body changes with time.
2. The slope of the displacement-time graph can determine the velocity of the body.
3. The slope of the velocity-time graph can determine the acceleration of the body.
4. The area under the velocity-time graph determines the displacement of the body.

Solved Problems Based on Graphical Representation of Equation of Motion

Q.1. A car is moving on a straight road at a uniform acceleration. The velocity of the car at different times is given below. Draw its speed-time graph and calculate its acceleration and also find out the distance covered by it in \(30\) seconds.

Time \(t\,\left( {\text{s}} \right)\)	\(0\)	\(10\)	\(20\)	\(30\)	\(40\)	\(50\)
Speed \(v\,\left( {\text{m/s}} \right)\)	\(5\)	\(10\)	\(15\)	\(20\)	\(25\)	\(30\)

Ans: The variation in the speed of the car moving on a straight road with time can be represented by the speed-time graph.

The slope of the speed-time graph gives the acceleration of the body.
Therefore, the acceleration \(\left( a \right)\) of the car is given by the following formula:
\(a = \frac{{{v_B} – {v_A}}}{{{t_B} – {t_A}}}\)
Where \({{v_A} = }\) Speed of the car at point \(A.\)
\( {v_B} = \)Speed of the car at point \(B\)
\({t_A} = \)Time at which the car is having speed \({v_A}\)
\({t_B} = \)Time at which the car is having speed \({v_B}\)
By putting the values of the speed and time in the above formula, we get:
\(a = \frac{{20 – 5}}{{30 – 0}} = 0.5\,{\text{m}}/{{\text{s}}^2}\)
The area enclosed by the speed-time graph gives the distance covered by the body.
Therefore, the distance \(\left( s \right)\) covered by the car in \(30\,{\text{s=}}\) area of \(OABC.\)
\(s = \) area of rectangle \(OADC + \)area of triangle \(ABD\)
\(s = OA \times OC + \frac{1}{2}\left({AD \times BD} \right)\)
\(s = 5 \times 30 + \frac{1}{2}\left({30 \times 15} \right)\)
\(s = 375\,{\text{m}}\)

Q.2. In the following figure, the position-time graph for an object at different times is shown. Calculate the speed of the object during \(C\) to \(D.\)

Ans: The change in the position of an object with time can be represented by the distance-time graph. In this graph, time is taken along the \(X\)–axis, and the distance is taken along the \(Y\)-axis. The slope of the distance-time graph gives the speed of the object.
The speed of the object during \(C\) to \(D\) will be given by:
\({v_{{\text{from}}\,\text{C}\,\text{to}\,\text{D}}} = \frac{{9 – 3}}{{9 – 7}} = \frac{6}{2} = 3\,{\text{cm/s}}\)

Summary

From this article, we can conclude that the graphical representation of motion can simplify the method of analysing and determining any quantity involved in the event of motion. The slope of the displacement-time graph represents the velocity of the object, and it also tells whether the object is moving with a constant velocity or with a variable velocity.

The slope of the velocity-time graph represents the acceleration of the object. The area enclosed by the velocity-time graph represents the displacement of the object.

FAQs on Graphical Representation of Motion

The most frequently asked questions on the graphical representation of the equation of motion are answered here:

Q.1: What are the two uses of graphical representation of motion?
A: The uses of the graphical representation of motion are as follows:
1. The graphical representation of motion helps to interpret the information about the distance, speed, and acceleration of an object at any instant of time immediately upon looking.
2. The velocity of a moving object can be easily determined by the displacement-time graph.

Q.2: How many types of graphs are there in motion?
A: There are two main types of graphs in motion, namely displacement-time graph and velocity-time graph.

Q.3: What is the graphical representation of linear motion?
A: The representation of the motion of a body moving along a straight line pictorially by a graph is called the graphical representation of linear motion. For example, the displacement-time graph of a car moving along a straight road with a uniform velocity in a straight line that is not parallel to the time axis.

Q.4: How important is a graphical representation in describing the motion of an object?
A: The graphical representation of the motion of an object not only makes the understanding of the nature of the move easier but also makes it simple to find the various parameters by looking at the graph itself. Some values of different parameters can be determined by observing the graph of the motion.

For example, if the graph of two objects which are moving with a constant velocity is plotted on the same distance-time graph, then the object for which the slope of the graph is maximum is moving with a greater speed.

Q.5: How do you find the velocity of an object from the graphical representation of motion?
A: The velocity of an object is defined as the displacement per unit time. The velocity of an object can be determined by the slope or gradient of the displacement-time graph for an object. The steeper the slope faster will be the object. The formula to find the velocity\(\left( v \right)\) of the object from the displacement-time graph is given below:

\(v = \frac{ { {s_2} – {s_1}}}{ { {t_2} – {t_1}}}\)
where, \({s_1}\) and \({s_2}\) are the displacements at time instant \({t_1}\) and \({t_2}\), respectively.