Graphs

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What is Graph?

There are so many things to learn about the purpose of the graph in Physics. A graph is the way of expressing the relationship between two quantities, out of which one alters as an after-effect from the other.

We need some basic information about some points related to graphs such as;

  1. What are the requirements of graphs?

We require graphs in physics as it is the most useful and powerful method of presenting the data. 

  1. How to design a graph?

We can plot both tables and graphs to represent the data, but graphs are quite easier to manipulate and interpret data than tables.

There are some terms used in graphs such as:

  • Independent variable, and 

  • Dependent variable. 

The Independent variable is known as the variable that is made to alter. 

The dependent variable is another variable that alters as a result of the change in the dependent variable.


Types of Graphs

Graph study is similar to that of kinematics; however, the only difference is that it is in graphical form.

We will be learning the three important types of graphs such as;

Displacement-time (d-t)

Velocity-time (v-t)

Acceleration-time (a-t)

We have learned a mathematical approach to speed, distance, velocity, and displacement. These graphs will help us to understand better about the motion. 

As per the physics, the observer should be able to interpret motion by visualizing the graph.


Displacement Time Graph

Have you encountered the term displacement? It can be defined as how far is the object from its initial point. 

In this graph, displacement is taken as the dependent variable and is represented on the y-axis. In the x-axis, the independent variable is represented as time.

This is also known as the Position-time graph. The three different plots for the Displacement-time graph are given below.

[Image will be Uploaded Soon]

The slope of this graph will always be equal to the velocity of the object at that particular time. The slope can be illustrated as;

S = \[\frac{displacement}{time}\] = \[\frac{y_{2} - y_{1}}{x_{2} - x_{1}}\] = \[\frac{Δd}{Δt}\] = velocity  

The following points are the takeaway obtained from the displacement-time graph

  1. Constant velocity is expressed by a straight line, where at the same time acceleration is expressed as a curved line.

  2. Velocity is equal to the slope.

  3. When an object is at rest, it implies that the slope is zero.

  4. The motion in a positive direction indicates a positive slope.

  5. The negative slope implies that the motion is in a opposite direction.


Velocity Time Graph

This helps to plot a graph between the velocities of an object relative to a certain point, with time. Time is plotted on the x-axis and velocity on the y-axis.

We know that velocity is the derivative of the distance function.

If we plot any velocity-time graph and want to know the total distance between the two points in time, we can find the area under that graph between those two points in time.

Generally if, v = f(t) 

s = \[\int_{t_{1}}^{t_{2}}\] f(t) d(t)

The slope can be illustrated as

Slope S = \[\frac{velocity}{time}\] = \[\frac{y_{2} - y_{1}}{x_{2} - x_{1}}\] = \[\frac{Δv}{Δt}\] = acceleration      

[Image will be Uploaded Soon]

We notice that the velocity-time graph’s slope is the definition of acceleration. So this graph can be illustrated that the slope is equal to the acceleration. 

  1. The following points are the takeaway obtained from the velocity-time graph;

  2. The steep slope indicates the spontaneous change in velocity.

  3. Shallow slope implies the dragging change in velocity.

  4. If the slope is positive, then acceleration will also be positive also.

  5. IF the slope is negative, then the acceleration will be negative.


Acceleration Time Graph

The acceleration-time graph illustrates that the acceleration is the dependent variable and is plotted on the y-axis, and time is plotted on the x-axis which is the independent variable.

The slope S can be illustrated as;

S = \[\frac{acceleration}{time}\] 

= \[\frac{y_{2} - y_{1}}{x_{2} - x_{1}}\]

= \[\frac{Δa}{Δt}\]

The slope of the graph of acceleration-time is called a jerk. Some points are acknowledged from the graph, these are as follows:

[Image will be Uploaded Soon]

  1. If the slope is zero, the motion will have a constant acceleration.

  2. The area under the graph implies the alteration in velocity.


Velocity Time Graph Examples

To draw velocity-time graphs, we are going to use three equations of motion.


Case 1 - Velocity-time Graph with Zero Acceleration (Constant Velocity):

[Image will be Uploaded Soon]

We can see in the diagram drawn above. This happens only when velocity is constant in the velocity-time graph where y-axis denoting velocity and x-axis denoting time.

We can clearly say that the velocity is constant (c) throughout the total time interval. The velocity does not vary with how much the time changes. 

In this situation, we have taken the positive for the initial velocity. The graph will be different if we change the velocity value into negative.


Case 2 - Velocity- time Graph at Constant Acceleration:

When the acceleration is constant (positive acceleration), and the initial velocity is zero of the particle, there will be a linear rise in velocity's particle as per the equation given by:

v = u + at

Since u = 0

So, v = at

[Image will be Uploaded Soon]

As shown in the graph, the velocity will increase linearly with time. The magnitude of acceleration can be obtained through the slope of the graph.


Case 3 - Velocity-time Graph with Increasing Acceleration:

When the acceleration is rising with time, the velocity-time graph will be plotted as a curve according to the equation;

v = u + at

Though, u = 0

v = at

[Image will be Uploaded Soon]

Though acceleration is a function of time, the graph will be a curve. It is so because acceleration is continuously rising with time, the magnitude of the slope is also rising continuously with time.

FAQ (Frequently Asked Questions)

Q1. How Do We Plot Graphs?

Ans: Graphs are created from data tables. Since most of the data collected by the scientists’ are quantitative, it is easy to plot tables and charts for easy use. The graph helps in the studies of mechanics.

Q2. How Do You Elaborate on the Motion on a Velocity-time Graph?

Ans: The velocity-time graphs are utilized to describe the motion of objects which are moving in a straight line. The direction of the object may change, such as forward to backward, left or right, and up or down. This graph is similar to the speed-time graphs.

Q3. What are the Utilizations of Graphs?

Ans: Graphs are common methods to illustrate as well as visualize the relationship between the data. In two dimensional graphs, it generally has two-axis, such as the x-axis and the y-axis. 

The circle graphs show the different parameters as per requirement.

Q4. Why are Graphs Not Complicated to Understand?

Ans: Graphs can represent numbers, words, measurements, or observations in a way that makes it easier to learn free from complications. It is crucial to make sure that the data you input in the graph are represented clearly and understandable.