### 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 of time is plotted against the corresponding time values, the graph turns out to be a straight line that passes through the origin (if the body’s motion is considered to start from rest), and is inclined to the x-axis (time-axis). Such a graph when plotted for a moving body can provide a lot of information about the motion of the body such as, the type of motion, velocity and acceleration of the body, and much more.

### Explanation:

**Slope of a graph:**

The slope is a measure of steepness of a graph. Consider two points A and B in the position-time graph of a body in a uniform motion and let their coordinates be (t_{1}, s_{1}) and (t_{2}, s_{2}) respectively as shown in the figure below.

The slope of this graph is given by,

Slope, m = BC/ AC

Or, **m = (s**_{2}** – s**_{1}**)/ (t**_{2}** – t**_{1}**)**

In fact, the **slope **of a s-t graph at a particular instant gives the **velocity** of the body at that instant.

This is because the slope is the ratio of the change in displacement and change in time, which is equal to the velocity of the body.

Or, **v = m = (s**_{2}** – s**_{1}**)/ (t**_{2}** – t**_{1}**)**

**Types of position time graphs:**

The position time graph may look very different for different kinds of motion. Some of them are as follows:

**Position time graph of a body in a uniform motion**

An object moving at a **uniform velocity** covers equal distances in equal intervals of time, as shown in the data table given below. Such a motion is called a **uniform motion**. Its **s-t graph **is a **straight line passing through the origin, **as shown in the graph below. The **slope** of the position-time graph of a body in uniform motion is **constant **throughout its motion. This indicates that the **velocity** of the body in such a motion is **constant** throughout its motion.

Therefore, the velocities of different bodies can be **compared** by looking at the steepness (slope) of their s-t graphs. For example, the graph shown below shows the s-t graph of two bodies in uniform motion at different velocities. It is easy to figure out from the graph that the velocity of the first body is greater than the second.

**Position time graph of a body in a uniform motion at a negative velocity**

An object’s velocity can be considered to be negative when it moves in the opposite direction w.r.t its initial direction of motion. As the body heads towards its initial point, its displacement keeps decreasing until it reaches its initial point, where the displacement becomes zero. The **slope** of the position-time graph of a body in uniform motion at a **negative velocity** (velocity opposite to the direction of motion) is **negative**. A sample data table and the corresponding graph are shown below is the s-t graph of a body moving at a constant negative velocity.

**Position time graph of a body at rest**

The **displacement** of a body at rest **remains the same** until it starts moving. Therefore, the s-t graph of a body at rest is a **straight line parallel to the x-axis** (time-axis) as shown in the figure below.

**Position-time graph of a body in a uniformly and positively accelerated motion**

The distance covered in a second by a body **keeps increasing uniformly **in a uniformly and positively accelerated motion. The data table and the corresponding s-t graph of such a motion is shown in the figure below. The **acceleration** of such a body remains constant and is directed along the direction of motion of the body throughout its motion. The acceleration of the body in the data table given below can be calculated to be **+2 m/s ^{2}**. The graph of such a motion is in the shape of a

**parabola**as shown below.

The slope of such a graph which is not a straight line can be found out by drawing a tangent at the desired point on the graph and calculating the slope of the tangent in the usual method. If tangents are drawn at several points, it can be seen that the **slope** of the above graph **keeps increasing** as we move away from the origin towards the right. This indicates that the **velocity **of the body **keeps increasing** as it moves.

**Position-time graph of a body in a uniformly and negatively accelerated motion**

The distance covered in a second by a body **keeps decreasing uniformly **in a uniformly and negatively accelerated motion. The data table and the corresponding s-t graph of such a motion is shown in the figure below. The **acceleration** of such a body remains constant and is directed **opposite to** the direction of motion of the body throughout its motion. The acceleration of the body in the data table given below can be calculated to be **-2 m/s**** ^{2}**. The graph of such a motion is in the shape of a

**parabola**as shown below.

If tangents are drawn at several points, it can be seen that the **slope** of the above graph **keeps decreasing** as we move away from the origin towards the right. This indicates that the **velocity **of the body **keeps decreasing** as it moves.

**Position-time graph of a body in a non-uniform motion.**

The velocity of a body in a non-uniform motion (except the uniformly accelerated/decelerated motion) lacks a pattern. Therefore, the slope of the **s-t** graph of such a body has **no pattern**. Thus, s-t graph of a non-uniform motion has **no definite shape**. A sample s-t graph of such a motion is shown below.

In the above s-t graph, the velocity of the body increases for some time and then starts to decrease.

## How are graphs of position vs time useful?

A lot of people have vague anxiety and a strong wish for the experience to end as soon as possible when they encounter graphs, just like they do when they visit the dentist. However, position graphs can be attractive and are a useful way to graphically depict a great deal of data about an object’s motion in a neatly condensed area.

**Problems**

- What kind of motion does the body undergo at every section of the graph shown below?

**Answer:**

A – Uniform motion in the positive direction

B – Rest

C – Uniform motion in a negative direction

D – Uniform motion in the negative direction

E – Uniform motion in the positive direction

- Indicate the type of slope (positive, negative, zero) for each section of the graph.

**Answer:**

A – Positive

B – Zero

C – Negative

D – Negative

E – Positive

- Calculate the velocity of the body during each section of the graph.

**Solution:**

**SECTION – A**

Consider the end-points on the s-t graph with coordinates (0, 0) and (10, 60).

Velocity = slope = (60 – 0)/ (10 – 0)

= 60/ 10

= **6 m/s**

**SECTION – B**

Consider the end-points on the s-t graph with coordinates (10, 60) and (15, 60).

Velocity = slope = (60 – 60)/ (15 – 10)

= 0/5

= **0 m/s**

**SECTION – C**

Consider the end-points on the s-t graph with coordinates (15, 60) and (30, 0).

Velocity = slope = (0 **–** 60)/ (30 **–** 15)

= **–** 60/ 15

= **–**** 4 m/s**

**SECTION – D**

Consider the end-points on the s-t graph with coordinates (30, 0) and (40, –40).

Velocity = slope = (–** **40 – 0)/ (40 – 30)

= – 40/ 10

= **– 4 m/s**

**SECTION – E**

Consider the end-points on the s-t graph with coordinates (40, – 40) and (55, 0).

Velocity = slope = (0 – (– 40))/ (55 – 40)

= 40/ 15

= **2.66 m/s**

### Summary

1. In the position-time graph, the distance/displacement of a body is plotted against time.

2. The slope of a graph is the measure of steepness of the graph.

3. The slope of a position-time graph at any instant is equal to the velocity of the body at that instant.

4. The position-time graph for a body in a uniform motion is a straight line inclined to the x-axis with a non-zero slope.

5. The position-time graph of a body in a uniformly accelerated motion is in the shape of a parabola.

## FAQs

- How is a position vs. time graph drawn?

Ans) By placing the motion’s time on the x-axis and the position on the y-axis, a position vs. time graph can be created.

- Why are graphs of position vs time important?

Ans) Graphs of position against time are crucial because they allow us to carefully study the trajectory and position of the object.

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