Velocity - Time Graphs

The velocity of an object is defined as the displacement (covered by it) per unit time in a particular direction.

• It is a vector quantity, i.e., it has both magnitude and direction.
• It is denoted by the symbol v, has an SI Unit of m/sec, and its dimensions are [LT^-1]

A Velocity-Time graph shows how an object's velocity changes over time when it's moving in a straight line. The graph has velocity on the vertical axis and time on the horizontal axis.

Depending upon the different types of velocities, for example, an object may have a constant velocity, increasing velocity, or decreasing velocity, it can easily be determined if the object is accelerating, decelerating, or not accelerating at all.

Key Concepts

1. The velocity–time graph provides two fundamental relationships in uniformly accelerated motion:

2. To draw velocity-time graphs, we will use the three equations of motion:

1. v = u + at
2. s = ut + 1/2(at²)
3. v² = u² + 2as

3. The following velocity–time graph shows how these concepts apply to an object moving in a straight line:

CASE-A [Uniform Acceleration]

Uniform acceleration means velocity changes by the same amount in equal time intervals, like a freely falling object, with constant acceleration.

Case A1 : V-t graph with Constant Velocity (Zero Acceleration)

Case A2 : V-t graph with Increasing Velocity (Constant Acceleration)

Case A3 : V-t graph with Decreasing Velocity (Constant Retardation)

CASE-B [Non-Uniform Acceleration]

Non-uniform acceleration means velocity changes by unequal amounts in equal time intervals, like a car in traffic, with changing acceleration.

Case B1 : V-t graph with Increasing Velocity (Increasing Acceleration)

Case B2 : V-t graph with Decreasing Velocity (Decreasing Acceleration)

Calculation of Distance and Displacement from V-t graph

To calculate distance and displacement from a velocity-time (v-t) graph, find the area under the curve.

• Displacement is the net area (positive areas minus negative areas)
• Distance is the total area (sum of all areas, treating negative areas as positive).

Both are found by breaking the graph into shapes (triangles, rectangles, trapezoids) and adding their individual areas. 

Illustration 1: Calculation of Distance and Displacement of the following graph from 0 to 6 seconds.

Related Articles

• Graphical Representation of Motion
• Distance-Time Graph
• Acceleration-Time Graph

Solved Examples:

Question 1: Draw the graph for an object starting at a velocity of 20m/sec and the final velocity becomes 0 m/sec at T=10th second. The object is decelerating in nature.

The graph will have a negative slope since the object is deceleration in nature.

Question 2: An object with an initial velocity of 15m/sec is decelerating and ultimately coming to a stop at T= 20 seconds, Find the Displacement covered by the object.

The displacement of the object can be found out by finding the area of the velocity-time graph. 

Area of the right angle triangle= Displacement of the object= \frac{1}{2} × Base × Height

Displacement= \frac{1}{2}× 20× 15

Displacement = 150 meters.

Question 3: Find the Acceleration of the object which is constant in nature, The graph for the object's velocity is given below.

The acceleration in the velocity-time graph is found out through the slope, the slope is the ratio of the value at y-axis and x-axis, solving this will easily give the value of acceleration of the object.

At time T= 8 sec, the value of the velocity of the object is 8 m/sec.

Hence, the slope of the graph at T= 10 sec will be,

Slope of the graph= Acceleration of the object = 8/10

Acceleration = 0.8 m/sec²

Question 4: A Ball is thrown upwards and then after reaching a particular height it come down as shown in the figure Draw the following graph for the above case:-

1. Distance - Time graph
2. Displacement - Time graph
3. Speed - Time graph
4. Velocity - Time graph
5. Acceleration -Time graph

Question 5: A Ball is dropped downwards and then after colliding with the ground it rebounds back as shown in the figure Draw the following graph for the above case:-

1. Distance - Time graph
2. Displacement - Time graph
3. Speed - Time graph
4. Velocity - Time graph
5. Acceleration -Time graph

Question 6: Draw the Following Velocity-time graph for the following motions with constant acceleration:-

1. Motion in positive direction with positive acceleration
2. Motion in positive direction with negative acceleration,
3. Motion in negative direction with negative acceleration,
4. Motion of an object with negative acceleration that changes direction at time t₁. Between times 0 to t, it moves in positive x - direction and between t, and t, it moves in the opposite direction.

Unsolved Questions

Q1. The velocity versus time graph of a body moving in a straight line is as shown in the figure below, Find

1. The distance covered by the body in 0 to 3s also
2. The acceleration of the body in 0 to 2s

Q2. The speed-time graph of a particle moving along a solid curve is shown below. The distance traversed by the particle from t = 0 to t = 3 is ?

Q3. The velocity (v) of a particle moving along x-axis varies with its position x as shown in figure. Find the acceleration (a) of particle varies with position (x)? Hint:(Find the Equation between a and x)