Graphs

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. 

What are Some of the Basic Learnings that we get from Graphs In Physics?

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. Both tables and graphs are used to represent the same information but graphs are a lot easier to read and interpret information better than tables.  A graph can show that the variables (dependent and independent) change according to some laws of physics. A graphical representation of any data can give a qualitative relation of the variables without the need to perform any further calculations.  

2. 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.

Let’s take a look at how to plot graphs for calculation: 

• Every graph should have a short, elaborative title (best to be mentioned at the top of each graph), detailing the units that are being measured. Both the axes should be clearly labeled with the variable titles and units.

• The label of the scale of each axis should be mentioned clearly. For example, "2 squares = 0.1 meters".

• The dots on a graph should not be connected, but rather they should be given a best-fit line or curve.

• The best-fit line should be drawn with a ruler (not by freehand), and should closely approximate the trend of all the data, not any single point or group of points.

• The best-fit curve should extend beyond the data points.

• From two points on the best fit curve, calculate the slope. The two points should be taken in such a manner that they are not close to each other. No data point should be used to calculate the slope.

• Calculate the slope of the graph, in the graph itself. Units should be included, and the value of the slope should be easily visible.

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): A displacement -time graph plots the distance of an object from a certain point. X-axis denotes time, while Y-axis denotes distance. 

Velocity-time (v-t): A velocity-time graph denoted the velocity of an object in motion from a certain point. X-axis denotes time, while Y-axis denotes velocity. 

Acceleration-time (a-t): The acceleration-time graph determines the change in velocity at regular intervals of time. 

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 vs Time Graph

Have you encountered the term displacement? It can be defined as how far is the object from its initial point.(the shortest distance between two points.)

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.

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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;

               \[Slope,\Theta =\frac{displacement}{time}=\frac{y_2-y_1}{x_2-x_1}\] 

Here, \[\frac{y_2-y_1}{x_2-x_1}\]=\[\frac{\Delta d}{\Delta t}\] = 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 the opposite direction.

Velocity vs Time Graph

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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.

  \[\Theta =\int_{t1}^{t2}f(t)d(t)\]

The slope can be illustrated as:

\[\Theta =\frac{velocity}{time}=\frac{y_2-y_1}{x_2-x_1}\]=\[\frac{\Delta v}{\Delta t}\]

Here, \[\frac{\Delta v}{\Delta t}\]=acceleration measured in ms⁻².

We notice that the velocity-time graph’s slope is the definition of acceleration. So this graph shows 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.

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):

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

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

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.

Acceleration vs Time Graph

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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;

             \[\Theta =\frac{acceleration}{time}\]

            Also, \[\Theta =\frac{y_2-y_1}{x_2-x_1}\]

So, slope \[\Theta =\frac{\Delta v}{\Delta t}\] 

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

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

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