Velocity Time Graphs – Definition, Examples

Key Concepts

1. Velocity-time graph

2. Slope of v-t graph = acceleration

3. Types of velocity-time graphs

4. Displacement from v-t graphs

What is Velocity Time Graph

A velocity time graph is a simple graph that shows you how the velocity of an object changes with time.

In this graph:

• Time is shown on the x-axis (horizontal line)
• Velocity is shown on the y-axis (vertical line)

It tells us:

• How fast an object is moving
• Whether the object is speeding up
• Whether the object is slowing down
• Whether the object is moving at the same speed
• Whether the object is not moving at all

You can see that: 

• If the velocity increases with time, the line goes upward
• If it decreases, the line goes downward
• If it stays the same, the line is straight and flat

It is a graph that shows velocity on one side and time on the other side. So we can clearly see how motion changes.

Introduction

The velocity of a body in a uniformly accelerated motion increases by equal amounts in equal intervals of time. This also indicates that it moves at a constant acceleration. When its velocity at different instants of time is plotted against the corresponding values of time, 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, acceleration, and displacement. 

How to Read a Velocity Time Graph

It is very easy to understand it if you follow these steps: 

Step 1: Look at the axes

First, you need to check the graph carefully: 

• The bottom line shows time
• The side line shows velocity

Time increases from left to right.

Step 2: Look at the shape of the line

If the line is:

• Straight and flat: The object is moving with constant velocity
• Straight and going up: The object is speeding up
• Straight and going down: The object is slowing down
• Curved upward: The object’s acceleration is increasing
• Curved downward: The object’s acceleration is decreasing

Step 3: Check the slope

• Steeper line: More acceleration
• Less steep line: Less acceleration
• Flat line: Zero acceleration

Step 4: Check the area under the graph

The area under the graph tells us the displacement.

We find its area using the formula for the shape it makes, if the graph makes a: 

• Triangle
• Rectangle
• Trapezium

That area gives the displacement.

So, how to read a velocity time graph becomes easy if we:

• Look at the axes
• Look at the slope
• Look at the area
• Look at the shape

Explanation:

Acceleration from a v-t graph: 

Acceleration in velocity time graph is found by calculating the slope of the graph. Consider two points A and B in the position-time graph of a body in a uniform motion and let their coordinates be (t1, v1) and (t2, v2), respectively, as shown in the figure below. 

The slope of this graph is given by, 

Slope, m = BC/ AC 

Or, m = (v2 – v1)/ (t2 – t1) 

In fact, the slope of a v-t graph at a particular instant gives the acceleration of the body at that instant.  

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

Or, v = m = (v2 – v1)/ (t2 – t1) 

Slope means: 

change in velocity ÷ change in time

• If the velocity increases quickly, the slope is big
• If the velocity increases slowly, the slope is small
• If velocity does not change, the slope is zero
• If the velocity decreases, the slope is negative

So, it is simply the slope of the line.

• Positive slope: Positive acceleration
• Negative slope: Negative acceleration
• Zero slope: No acceleration

Types of velocity-time graphs: 

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

1. Velocity-time graph of a body in a uniform motion: 

An object moving at a uniform velocity covers equal distances in equal intervals of time. This is because its velocity remains constant throughout the motion. Such a motion is called a uniform motion. Its v-t graph is a straight line parallel to the x-axis, as shown in the graph below. The slope of the velocity-time graph of a body in uniform motion is zero. This indicates that the acceleration of the body in such a motion is zero. 

2. Velocity-time graph of a uniformly and positively accelerating body: 

An object in a uniformly and positively accelerated motion increases its velocity by equal amounts in equal intervals of time. This is because its acceleration remains constant throughout its motion. The v-t graph of such a motion, when plotted, turns out to be a straight line passing through the origin (if the body starts from rest) with a positive slope. The positive and constant slope indicates the positive and constant acceleration of the body. 

3. Velocity-time graph of a uniformly and negatively accelerating body: 

An object in a uniformly and negatively accelerated motion decreases its velocity by equal amounts in equal intervals of time. This is because its acceleration remains constant but negative throughout its motion. The v-t graph of such a motion, when plotted, turns out to be a straight line with a negative slope. The negative and constant slope indicates the negative and constant acceleration of the body. 

Comparing accelerations: 

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

4. Velocity-time graph of a body at rest: 

The velocity of a body at rest is zero until it starts moving. Therefore, the v-t graph of a body at rest is a straight line coinciding with the x-axis (time-axis), as shown in the figure below.   

5. Velocity-time graph of a body with increasing acceleration: 

In this type of motion, the magnitude of the acceleration of a body keeps increasing with time. The plot of velocity and time for such a motion is curved in a way shown in the figure below. The graph would start from the origin if the body was initially at rest. Else, it may start from a point on the x or y-axis or any other point on the cartesian plane. The slope of the v-t graph of a body moving with increasing acceleration keeps increasing as the steepness of the graph keeps increasing. 

6. Velocity-time graph of a body with decreasing acceleration: 

In this type of motion, the magnitude of the acceleration of a body keeps decreasing with time. The plot of velocity and time for such a motion is curved in a way shown in the figure below. The graph would start from the origin if the body was initially at rest. Else, it may start from a point on the x or y-axis or any other point on the cartesian plane. The slope of the v-t graph of a body moving with decreasing acceleration keeps decreasing as the steepness of the graph keeps decreasing. 

Displacement from a velocity-time graph: 

The displacement covered by a body is equal to the area under its velocity-time graph. In the velocity-time graph given below, let the coordinates of A and B be (t, v) and (t, 0), respectively. 

Total displacement covered by the body = the area under the v-t graph (OA)  

                                                                           = area of the triangle OAB. 

Therefore,  

Total displacement = area(triangle OAB) 

                              = (1/2) x AB x OB 

                              = (1/2)vt 

The v-t graph might also look different in other cases. Or we might need to find the displacement of the body in a section of the graph. Even in those cases, the area under the graph gives the displacement of the body. Here is another example. 

Displacement = area under the v-t graph  = area (trapezium OABD) 

Or, s = area (ABC) + area (OACD) 

Or, s = (1/2) x AC x BC + AC x CD 

Or, s = (1/2) t2(v2 – v1) + t2v1 

Or, s = t2 [(1/2) (v2 – v1) + v1] 

If the displacement of a portion of the motion is required, then the area under the corresponding portion of the v-t graph should be calculated. 

Problems 

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

Answer: 

1. Non-uniformly accelerated motion  
2. Uniformly accelerated motion  
3. Uniform motion  
4. Uniformly decelerated motion  
5. Uniformly accelerated motion  
6. Non-uniformly decelerated motion  

2.Identify the type of motion (uniform, non-uniform, uniformly, and non-uniformly accelerated/ decelerated motion) the body is undergoing by analyzing every section of the velocity-time graph shown below.​ 

Answer: 

1. Non-uniformly accelerated motion  
2. Uniformly accelerated motion  
3. Uniform motion  
4. Uniformly decelerated motion  
5. Uniformly accelerated motion  
6. Non-uniformly decelerated motion  

3.The v-t graph of a body’s motion is shown in the figure below. Calculate its total displacement. 

Solution: 

Total displacement = area (OABD) + area (BDC) 

                              = AB BD + (1/2) BD DC 

                              = 3 x 6 + (1/2) x 6 x (7 – 3) 

                              = 18 + (1/2) x 6 x 4 

                              = 18 + 12 

                              = 30 m 

4.The v-t graph of a body’s motion is shown in the figure below. Calculate the acceleration of the body in each section of the graph. In which section does the body have the greatest acceleration? 

Solution: 

SECTION – A 

Consider the end-points on the v-t graph with coordinates (0,0) and (25, 50). 

Acceleration = slope = (50 – 0)/ (25 – 0) 

                                 = 50/ 25 

                                 = 2 m/s2 

SECTION – B 

Consider the end-points on the v-t graph with coordinates (25, 50) and (60, 50). 

Acceleration = slope = (50 – 50)/ (60 – 25) 

                                 = 0 

SECTION – C 

Consider the end-points on the v-t graph with coordinates (60, 50) and (70, 80). 

Acceleration = slope = (80 – 50)/ (70 – 60) 

                                 = 30/ 10 

                                 = 3 m/s2 

The body has the greatest acceleration in section C. 

Graph of Velocity vs Time

This is another name for the velocity time graph. Both mean the same thing.

In the graph of velocity vs time:

• Time is on the horizontal axis
• Velocity is on the vertical axis

This graph helps us understand motion in a very clear way.

It shows:

• When the object starts moving
• When it stops
• When it moves faster
• When it moves more slowly
• How long has it moves

If the graph passes through the origin, it means the object started from rest. If it stays on the time axis, the object is not moving. If it goes below the axis, the velocity is negative and the object is moving in the opposite direction.

The graph of velocity vs time is very useful in physics. It is because it gives you full information about motion in one simple picture.

Summary

1. In the velocity-time graph the velocity (y-axis) of a body is plotted against the time (x-axis) of a body.

2. The v-t graph for a motion with uniformly increasing velocity is a straight line with a positive slope.

3. The v-t graph for a motion with a uniformly decreasing velocity is a straight line with a negative slope.

4. The acceleration of a moving body is equal to the slope of its v-t graph.

5. Greater the slope of a v-t graph, greater the acceleration of the body.

6. The v-t graph of a body at rest coincides with die x-axis.

7. The v-t graph of a body with an increasing/decreasing acceleration is a curve whose slope keeps increasing/decreasing as the time increases.

8. The displacement of a body is equal to the area under the v-t graph of a body.

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