**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:

- Position-time graph

- 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.

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.

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

**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.

### 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**.

**Examples:**

**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?**

**Solution:**

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

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.

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

**Solution:**

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

- Uniform motion

- Stationary

- Non-uniform motion

- 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.

**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/s^{2}. The data table and the velocity-time graph of a ball dropped from the top of a high-rise building are given below.

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.

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/s^{2}. The data table and the velocity-time graph for a stone thrown upwards with a velocity of 60.8 m/s are given below.

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.

### 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**.

**Examples:**

**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.

**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.**

**Solution:**

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

- Positive non-uniform acceleration

- Positive uniform acceleration

- Zero acceleration

- Negative uniform acceleration

- Positive uniform acceleration

- Negative non-uniform acceleration

- 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:

- Non-uniformly accelerated motion

- Uniformly accelerated motion

- Uniform motion

- Uniformly decelerated motion

- Uniformly accelerated motion

- 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.

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