Need Help?

Get in touch with us

bannerAd

Graphing Motion

Aug 19, 2022
link

Key Concepts

  • Graphing Motion
  • Position-time graphs
  • Velocity- time graphs

Introduction: 

We usually create a data table to present the numerical values of different physical quantities involved in the motion of a body. However, a data table makes it quite tedious to analyze the dependence of one physical quantity (such as velocity, displacement, etc.) on another (such as time). Plotting the same data in a graph sheet makes it much easier to analyze the dependence of a physical quantity on another. Thus, graphing is a convenient way of representing the motion of a body. 

Explanation: 

There are two types of graphs that we will see in this section. They are as follows: 

  1. Position-time graph 
  1. Velocity-time graph 

Position-Time Graphs 

In position-time graphs, the position of an object is plotted against time. It shows how the displacement of a body changes with time. The displacement is plotted along the y-axis, and the time is plotted along the x-axis. The position-time graph for each type of motion appears different from another. 

Based on the type of motion, the position-time graphs are of the following types: 

Position-time graph for a uniform motion: 

A body is said to be in a uniform motion if it covers equal distances in equal intervals of time. The data table of a car undergoing uniform motion and the corresponding position-time graph are given below. 

parallel
 (a) Data table
(b) and
 (c) position-time graph for a uniform motion

The position-time graph for a uniform motion is a straight line passing through the origin. 

Position-time graph for a stationary body: 

A body is said to be stationary when its displacement is zero. Suppose a body moves a certain distance (say 600 m) and then stops. In that case, its position-time graph would look like the one shown in figure 4.2 below. 

Position-time graph of a stationary body

The position-time graph of a stationary body is a straight line parallel to the x-axis (time axis). 

parallel

Position-time graph for the non-uniform motion: 

A body is said to be in a non-uniform motion if it covers unequal distances in equal intervals of time. The position-time graph of a body undergoing a non-uniform motion is given in figure 4.3 below. 

 Position-time graph of a non-uniform motion

Comparing velocities 

Velocity is given by the steepness of the position-time graph of a motion. The steeper the graph, the larger the velocity of the body. Figure 4.4 below shows the position-time graphs of two buses A and B, moving with uniform velocities of their own. Here, the graph of A is steeper than B. This indicates that A moves with a greater velocity than B

Comparing velocities

Examples: 

  1. Alen starts from the point P and moves in the rectangular path shown below. He takes 2 minutes to travel each side. Fill in the data table, plot a position-time graph for his motion, and check whether the motion is uniform or non-uniform? 
 Example 1

Solution: 

The data table for Alen’s motion is given below. 

Position-time graphs for example 1
Position-time graphs for example 2

Alen’s motion is uniform in each arm of the rectangular path. In comparison, it is overall non-uniform, as he covers unequal distances in equal intervals of time. 

  1. Identify the type of motion the body is undergoing by analyzing every section of the graph shown below. 

 Example 2

Solution:

The body in the graph is undergoing the following types of motion: 

  1. Uniform motion 
  1. Stationary 
  1. Non-uniform motion 
  1. The figure below shows the position-time graph of a moving car. It has four sections: A, B, C, and D. Rank these sections in the ascending order of velocity.
 Example 3

Solution: 

D>B>A>C 

A = rank 3 

B = rank 2  

C = rank 4 

D = rank 1 

Velocity-Time Graphs 

In velocity-time graphs, the velocity of an object is plotted against time. It shows how the velocity of a body changes with time. The velocity is plotted along the y-axis, and the time is plotted along the x-axis. The velocity-time graph for each type of motion (acceleration) appears different from another. 

Based on the type of motion (acceleration), the velocity-time graphs are of the following types: 

Velocity-time graph for a uniformly accelerated body: 

A body is said to be in a uniformly accelerated motion when its velocity increases by equal amounts in equal intervals of time. A freely falling body undergoes a uniformly accelerated motion. Its velocity starts from 0 and increases by 9.8 m/s every second due to gravity. Therefore, it has a uniform acceleration of 9.8 m/s2. The data table and the velocity-time graph of a ball dropped from the top of a high-rise building are given below. 

Velocity-time graph for a uniformly accelerating body 1
Velocity-time graph for a uniformly accelerating body 2
Velocity-time graph for a uniformly accelerating body 3

The velocity-time graph of a uniformly accelerating body is a straight line passing through the origin. 

Velocity-time graph for a uniform motion (acceleration = 0) 

A body is said to be in uniform motion when its velocity does not change with time. Thus, such a body does not accelerate. A car moving with a constant velocity is an example of uniform motion.

data table and velocity-time graph are given below. 

 Velocity-time graph for a uniformly moving body 1
 Velocity-time graph for a uniformly moving body 2
 Velocity-time graph for a uniformly moving body 3

The velocity-time graph for a body in a uniform motion is a straight line parallel to the x-axis (time axis). 

Velocity-time graph of a uniformly decelerating body: 

A body is said to be uniformly decelerating if its velocity keeps decreasing equally in equal intervals of time. A body thrown upwards is an example of a uniformly decelerating body. Its velocity keeps decreasing by 9.8 m/s every second due to gravity until it becomes zero. Thus, its acceleration is 9.8 m/s2. The data table and the velocity-time graph for a stone thrown upwards with a velocity of 60.8 m/s are given below. 

Velocity-time graph for a uniformly decelerating body 1
Velocity-time graph for a uniformly decelerating body 2
Velocity-time graph for a uniformly decelerating body 3

The velocity-time graph for a uniformly decelerating body is a straight line with negative steepness (slope). 

Velocity-time graph of a non-uniformly accelerating and non-uniformly decelerating body: 

A non-uniformly accelerating body accelerates unequally in equal intervals of time. In contrast, a non-uniformly decelerating body decelerates (negatively accelerates) unequally in equal intervals of time. The velocity-time graphs are curved lines, as shown below. 

Velocity-time graph for (a) a non-uniformly accelerated body
(b) non-uniformly decelerating body] 

Comparing accelerations: 

The acceleration of a body is indicated by the steepness of the velocity-time graph of a body. Steepness is mathematically known as the slope. The steeper the graph, the greater the acceleration of the body. The figure below shows the velocity-time graphs of two trucks A and B, moving with uniformly increasing velocities. Here, the graph of A is steeper than B. Thus, A accelerates more than B

Comparing accelerations

Examples: 

  1. A biker starts from rest and reaches a velocity of 10 m/s in 10 seconds. She then slows down for 10 seconds until she reaches a velocity of 6 m/s. For the next 10 seconds, she accelerates uniformly and reaches a velocity of 14 m/s. After 10 more seconds, her velocity reaches 5 m/s. She stops after the next 10 seconds. Fill in the data table showing the velocity and time columns and plot a velocity-time graph for her motion. 

Solution: 

The data table can be filled as shown below, and the graph is also shown below. 

 Solution of example 1 
 Solution of example 2
 Solution of example 3

  1. Identify the type of acceleration (positive, negative, zero, non-uniform) the body is undergoing by analyzing every section of the velocity-time graph shown below. 
 Example 2

Solution: 

The body in the graph is undergoing the following types of acceleration: 

  1. Positive non-uniform acceleration 
  1. Positive uniform acceleration 
  1. Zero acceleration 
  1. Negative uniform acceleration 
  1. Positive uniform acceleration 
  1. Negative non-uniform acceleration 
  1. Identify the type of motion (uniform, non-uniform, uniformly, and non-uniformly accelerated/ decelerated) the body is undergoing by analyzing every section of the velocity-time graph shown above in figure 4.14 of example 2. 

Solution: 

The body in the graph is undergoing the following types of motion: 

  1. Non-uniformly accelerated motion 
  1. Uniformly accelerated motion 
  1. Uniform motion 
  1. Uniformly decelerated motion 
  1. Uniformly accelerated motion 
  1. Non-uniformly decelerated motion 

 

Summary:

  • Graphical representation of motion makes it convenient to analyze the motion of the body.
  • In the position-time graph, the displacement (y-axis) of a body is plotted against the time (x-axis) of a body.
  • The position-time graph for a uniform motion is a straight line passing through the origin.
  • The position-time graph for a non-uniform motion is a curved line.
  • The steeper the position-time graph, the larger the velocity.
  • In the velocity-time graph, the velocity (y-axis) of a body is plotted against the time (x-axis) of a body.
  • The velocity-time graph for a motion with uniformly increasing velocity is a straight line with positive steepness (slope).
  • The velocity-time graph for a motion with uniformly decreasing velocity is a straight line with negative steepness (slope).
  • The velocity-time graph for a non-uniformly accelerated motion is a curved line.
  • The steeper the velocity-time graph, the larger the acceleration.

Comments:

Related topics

Define Position Time Graph and its Types

Key Concepts • Slope of a graph • Position time graph • Slope of s-t graph = Velocity • Types of position time graphs Introduction An object in a uniform motion covers equal distances in equal intervals of time. This also indicates that it moves at a constant velocity. When its position at different instants […]

Read More >>

Magnetic Field Lines: Definition, Explanation and Q&A

Key Concepts Magnetic Field Magnetic Field Lines properties of magnetic field lines Uniform and non uniform magnetic lines Introduction Two magnets when placed close to each other attract and stick to each other. However, if we go on increasing the distance between them, the attraction between them reduces gradually to such an extent that they […]

Read More >>

The Life Cycles of Stars: Meaning and Example

Key Concepts Stars Analysis of starlight Composition of stars Stars’ temperature Size and mass of stars Stages of life cycle of a star Introduction Stars are huge, shining balls of extremely hot gas (known as plasma) in space. The Sun is our nearest star. During the nighttime, many other stars are visible to the naked […]

Read More >>

Mirror Formula

Key Concepts New cartesian sign convention Mirror formula Solving problems using the mirror formula Introduction When dealing with the reflection of light by spherical mirrors mathematically, a set of sign conventions is followed, called the New Cartesian Sign Convention. According to this convention, the pole of a spherical mirror is taken as the origin and […]

Read More >>

Other topics