An Introduction to Motion
When the position of an object changes with time with respect to some reference point, we can say that the object is in motion. The reference point is the point from which the change in position of an object is observed. Graphs offer an easy method to showcase the relationship between two physical quantities.
Importance of Graphs in Motion
Graphs provide us a convenient method to describe some basic information about a variety of events. The nature of the graph can be studied with the help of the slope. The slope is the inclination that the graph makes with the x-axis. It can also be found out by taking the ratio of change in the y-coordinates to change in the x-coordinates. Graphical representation of motion generally uses graphs and with the help of graphs, we can represent the motion of an object. In the graphical representation of motion, we should take the dependent quantity along the y-axis and the independent quantity along the x-axis.
Distance-Time Graph
Position of an object with time can be represented by a distance-time graph. The slope of the distance-time graph gives us the speed of an object. Distance-time graphs can be made by considering two situations, either the object moves with a uniform speed or with non-uniform speed. First, we will consider the case of uniform speed. If an object is moving with a uniform speed, we can say that it covers equal distances in equal time intervals. So, we can say that the distance is directly proportional to the time taken.
Suppose, the object covers 10m in the first 5 seconds, it covers another 10m in the next 5 sec and so on. If we mark these points on the graph, the result that we get would be like this:
Distance - Time Graph for Uniform Motion
Here, we can see a straight line passing from the origin. Thus, for a uniform speed, the graph of distance travelled against time is a straight line. We can also determine the speed of an object from the distance-time graph. The speed of an object can be determined by calculating the slope of the distance-time graph.
We can plot the distance-time graph for accelerated motion as well. Below is the example of the distance-time graph for non-uniform motion. The nature of this graph shows non-linear variation of the distance travelled by the object with time. So, the below graph shows the motion of an object with non-uniform speed.
Distance - Time Graph for Non-Uniform Motion
Displacement-Time Graph
Displacement is the shortest distance between two points. Displacement-time graph gives us the information of velocity of an object. The rate at which displacement varies with time is known as velocity and the slope of the displacement-time graph gives us the velocity of an object. So, if this graph gives us a straight line which is parallel to the x-axis, then we can say that object is at rest.
Velocity-Time Graph
The rate at which velocity varies with time is called acceleration and the slope of this graph gives us the information of acceleration of an object. The variation in velocity with time for an object moving in a straight line can be described by a velocity-time graph.
Velocity - Time Graph
Here, time and velocity are represented along the x-axis and y-axis, respectively. The graph shows a straight line which is parallel to the x-axis. That means the object has uniform velocity.
Solved Examples
1. An object is moving along a circular path of radius 7 cm. What is the distance of an object when it completes half revolution.
Ans: Given, radius $r = 7\,cm$
The object is moving along a circular path. We have to calculate the distance travelled by this object when it completes half revolution.
The circumference of the circle is $2\pi r$.
So, here, its distance travelled by it would be $\pi r$.
The distance of an object would be 22 cm when it completes half revolution.
2. A car decreases its speed from $22\,\dfrac{m}{s}$ to $16\,\dfrac{m}{s}$ in 5 sec. Find the acceleration of the car.
Ans: Given, initial speed of the car is $u = 22\,\dfrac{m}{s}$
Final speed of the car $v = 16\,\dfrac{m}{s}$
Time taken $t = 5\sec $
We will use the relation $a = \dfrac{{v - u}}{t}$ to solve this
$\therefore a = \dfrac{{16 - 22}}{5} = - 1.2\,\dfrac{m}{{{s^2}}}$
So, the acceleration of the car is $a = - 1.2\,\dfrac{m}{{{s^2}}}$
Interesting Facts
Displacement can be defined from the area under the velocity-time graph.
The displacement of an object is proportional to the square of time, then we can say that the object moves with uniform acceleration.
A speedometer is a good example of instantaneous speed.
Conclusion
Motion can be defined with the help of graphs. We use graphs to describe the motion of an object. In a line graph, we can represent one physical quantity like distance or velocity with another quantity such as time. The different types of graphs for motion help us to understand the speed, velocity, acceleration, and also behaviour of an object.
FAQs on Graphical Representation of Motion
1. What are the main types of graphs used to represent motion as per the Class 9 syllabus?
The three main types of graphs used to describe the motion of an object in a straight line are:
- Distance-Time Graph: Shows how the distance travelled by an object changes with time. Its slope gives the speed.
- Displacement-Time Graph: Represents the change in an object's position (displacement) over time. Its slope gives the velocity.
- Velocity-Time Graph: Illustrates how the velocity of an object changes with time. Its slope gives acceleration, and the area under it gives displacement.
2. What information does the slope of a distance-time graph provide?
The slope of a distance-time graph represents the speed of the object. A steeper slope indicates a higher speed, while a gentle slope means a lower speed. A horizontal line, which has a zero slope, indicates that the object is stationary (not moving).
3. What information can be obtained from a velocity-time graph?
A velocity-time graph provides two crucial pieces of information about an object's motion:
- The slope of the graph gives the acceleration. A positive slope means positive acceleration, a negative slope means negative acceleration (retardation), and a zero slope means constant velocity.
- The area under the graph represents the displacement of the object during that time interval.
4. How do you graphically distinguish between an object moving with uniform velocity and uniform acceleration?
You can distinguish them by observing their respective velocity-time graphs:
- Uniform Velocity: The velocity-time graph is a straight horizontal line parallel to the time axis. This shows that the velocity is constant and acceleration is zero.
- Uniform Acceleration: The velocity-time graph is a straight line with a constant, non-zero slope. The line is inclined to the time axis, indicating that velocity is changing at a constant rate.
5. Why is the graphical representation of motion an important tool in Physics?
Graphical representation is important because it provides a powerful visual method to understand and analyse motion. Its key advantages are:
- It offers a quick visual summary of the nature of motion (e.g., uniform, accelerating, at rest).
- It allows for the easy derivation of physical quantities like speed, velocity, acceleration, and displacement without complex calculations.
- It makes comparing the motion of two or more objects simple and intuitive.
6. Can a distance-time graph have a negative slope? Explain the reasoning.
No, a distance-time graph cannot have a negative slope. Distance is a scalar quantity that measures the total path length covered, and it can never decrease for a moving object. A negative slope would imply that the total distance is reducing, which is physically impossible. However, a displacement-time graph can have a negative slope, which correctly indicates that the object is moving back towards its starting point.
7. How is the total displacement of an object calculated from its velocity-time graph?
The total displacement of an object is calculated by finding the net area under its velocity-time graph for a specific time interval. You can calculate this by breaking the area into simple geometric shapes (rectangles, triangles). Any area above the time axis represents positive displacement, while any area below it represents negative displacement. The net displacement is the sum of these areas.
8. What does a curved line on a distance-time graph signify about an object's motion?
A curved line on a distance-time graph signifies non-uniform motion, specifically that the object is undergoing acceleration. Since the slope of a distance-time graph represents speed, a continuously changing slope (a curve) indicates that the speed is changing. If the curve gets steeper, the object is speeding up (accelerating).
9. What would the velocity-time graph for a ball thrown vertically upwards look like, from the moment it is thrown until it falls back to the starting point?
The velocity-time graph would be a single straight line with a constant negative slope. It starts at a maximum positive velocity, decreases linearly to zero (at the highest point), and then becomes increasingly negative as it falls back down. The constant negative slope represents the constant downward acceleration due to gravity (g).
