__What is called Displacement Time Graph?__

Characteristics of motion of a moving body can be quickly understood with the aid of graphs. For this purpose, the position, velocity and acceleration of a moving body are represented in vertical axis and time is represented on horizontal axis. These are called displacement time graph, velocity time graph and acceleration time graph etc.

*Therefore, displacement time graph is a graphical representation of motion of a moving body in which time is plotted in x-axis and displacement is plotted in y-axis.*

- In displacement time graph,
*displacement*is plotted along Y axis and time is plotted along X axis. - In velocity time graph,
*velocity*is plotted along Y axis and time is plotted along X axis. - In acceleration time graph,
*acceleration*is plotted along Y axis and time is plotted along X axis.

Displacement time graphs are important tool to understand the characteristics of motion. These are of following types –

- Displacement time graph of body at
*rest.* - Displacement time graph of body moving with
*uniform motion*. - Displacement time graph of body moving with
*uniform acceleration*. - Displacement time graph of body moving with
*retardation*.

### 1. Displacement time graph for a body at Rest

Consider about the figure showing graph of a body at rest –

- At time ( t_0 ) , the initial position of body is at point ( x_0 ) . Also at times ( t_1 ) and ( t_2 ) , the position of body is still at ( x_0 ) . This shows that, position of body is not changing with time.
- Hence, displacement – time graph AB is a straight line parallel to time axis.
- The slope of the displacement – time is zero. This represents zero velocity of body.
- The
*displacement*of body is not changing with time. Hence, the graph represents a body which is in rest.

### 2. Displacement time graph for Uniform Speed

Consider about the figure showing graph of a body moving with uniform speed –

- Displacement – time graph AB is a straight line with constant slope.
- At time ( t_0 ) , the initial position of body is at point ( x_0 ) . Also at times ( t_1 ) and ( t_2 ) , the positions of body are at ( x_1 ) and ( x_2 ) . This shows that, position of body is changing with time.
- Hence, the body is in motion.
- Displacement – time line ( AB ) is inclined to time axis. It has a constant slope.

Slope of the graph is –

\tan \theta = \left ( \frac {QR}{PR} \right ) = \left ( \frac {x_2 - x_1}{t_2 - t_1} \right )

Or, \quad \tan \theta = \left ( \frac {\text {Displacement}}{\text {Time}} \right ) = \text {Velocity}

- Therefore, slope of the graph represents the velocity of body.
- Constant slope indicates that the body is moving with uniform speed or velocity. Hence, acceleration acting on the body is zero.

### 3. Displacement time graph for Uniform Acceleration

Consider about the figure showing of a body moving with uniform acceleration –

- Displacement – time graph OB is a curve with increasing slope.
- In equal time intervals, displacements are increasing. Because, time intervals ( t_3 - t_2 ) and ( t_2 - t_1 ) are equal, but displacement ( x_3 - x_2 ) > ( x_2 - x_1 ) .
- Slope of the graph at point ( P ) is \quad \tan \theta_{1} = \left ( \frac {x_2 - x_1}{t_2 - t_1} \right ) .
- Slope of the graph at point ( Q ) is \quad \tan \theta_{2} = \left ( \frac {x_3 - x_2}{t_3 - t_2} \right ) .

But, \quad ( t_2 - t_1 ) = ( t_3 - t_2 ) \quad and \quad ( x_3 - x_2 ) > ( x_2 - x_1 )

Therefore, \quad ( \tan \theta_{2} > \tan \theta_{1} ) .

- Thus, slope of the graph is increasing. Therefore, it is representing an accelerated motion of a body.
- Curve ( OB ) is a parabola which bends upwards.

### 4. Displacement time graph of Retardation

Consider about the figure showing graph of a body moving with uniform retardation –

- Displacement – time graph OB is a curve with decreasing slope.
- In equal time intervals, displacements are decreasing. Because, time intervals ( t_3 - t_2 ) and ( t_2 - t_1 ) are equal, but displacement ( x_3 - x_2 ) < ( x_2 - x_1 ) .
- Slope of the graph at point ( P ) is \quad \tan \theta_{1} = \left ( \frac {x_2 - x_1}{t_2 - t_1} \right ) .
- Slope of the graph at point ( Q ) is \quad \tan \theta_{2} = \left ( \frac {x_3 - x_2}{t_3 - t_2} \right ) .

But, \quad ( t_2 - t_1 ) = ( t_3 - t_2 ) \quad and \quad ( x_3 - x_2 ) < ( x_2 - x_1 )

Therefore, \quad ( \tan \theta_{2} < \tan \theta_{1} ) .

- Thus slope of the graph is decreasing. Therefore, it is representing a retarded motion of a body.
- Curve ( OB ) is a parabola which bends downwards.

__Velocity Time Graph__

In velocity – time graph, velocity is plotted along Y axis and time is plotted along X axis.

These are of following types –

- Velocity time graph of body moving with
*uniform velocity.* - Velocity time graph of body moving with
*uniform acceleration.* - Velocity time graph of body moving with accelerated motion.
- Velocity time graph of body moving with retardation motion.

### 1. Velocity time graph of Uniform Velocity

Consider about the figure showing graph of a body moving with uniform velocity –

- At time ( t_0 ) , the initial velocity of body is ( v_0 ) . Also at times ( t_1 ) and ( t_2 ) , the velocities of body are ( v_0 ) . This shows that, velocity of body is not changing with time.
- Therefore, velocity time graph AB is a straight line parallel to time axis.
- The slope of the velocity time line ( AB ) is zero. This represents zero acceleration of body.
- The velocity of body is not changing with time.
- Hence, this graph represents a body moving with
*uniform velocity*.

### 2. Velocity time graph of Uniform Acceleration

Consider about the figure showing graph of a body moving with uniform acceleration –

- Velocity time graph AB is a straight line with constant slope.
- At time ( t_0 ) , the initial velocity of body is ( v_0 ) . Also at times ( t_1 ) and ( t_2 ) , the velocities of body are ( v_1 ) and ( v_2 ) . This shows that, velocity of body is changing with time.
- Hence, the body is in acceleration.
- Velocity – time line ( AB ) is inclined to time axis. It has a constant slope.

Slope of graph is –

\tan \theta = \left ( \frac {QR}{PR} \right ) = \left ( \frac {v_2 - v_1}{t_2 - t_1} \right )

Or, \quad \tan \theta = \left ( \frac {\text {Change in velocity}}{\text {Time}} \right ) = \text {Acceleration}

- Therefore, slope of the graph represents the acceleration of body.
- Constant slope indicates that, velocity of body is increasing with uniform rate.
- Hence, body is moving with
*uniform acceleration*.

### 3. Velocity time graph of Accelerated motion

Consider about the figure showing graph of a body moving with accelerated velocity –

- Velocity time graph AB is a curve with increasing slope.
- In equal time intervals, change in velocity is increasing. Because, time intervals ( t_3 - t_2 ) and ( t_2 - t_1 ) are equal, but change in velocity ( v_3 - v_2 ) > ( v_2 - v_1 ) .
- Slope of the graph at point ( P ) is \quad \tan \theta_{1} = \left ( \frac {v_2 - v_1}{t_2 - t_1} \right ) .
- Slope of the graph at point ( Q ) is \quad \tan \theta_{2} = \left ( \frac {v_3 - v_2}{t_3 - t_2} \right ) .

But, \quad ( t_2 - t_1 ) = ( t_3 - t_2 ) \quad and \quad ( v_3 - v_2 ) > ( v_2 - v_1 )

Therefore, \quad ( \tan \theta_{2} > \tan \theta_{1} ) .

- Slope of the graph is increasing. Therefore, it is representing an accelerated velocity of a body.
- Curve ( OB ) is a parabola which bends upwards.

### 4. Velocity time graph of Retarded motion

Consider about the figure showing graph of a body moving with retarded velocity –

- Velocity time graph OB is a curve with decreasing slope.
- In equal time intervals, velocity is decreasing. Because, time intervals ( t_3 - t_2 ) and ( t_2 - t_1 ) are equal, but change in velocity ( v_3 - v_2 ) < ( v_2 - v_1 ) .
- Slope of the graph at point ( P ) is \quad \tan \theta_{1} = \left ( \frac {v_2 - v_1}{t_2 - t_1} \right ) .
- Slope of the graph at point ( Q ) is \quad \tan \theta_{2} = \left ( \frac {v_3 - v_2}{t_3 - t_2} \right ) .

But, \quad ( t_2 - t_1 ) = ( t_3 - t_2 ) \quad and \quad ( v_3 - v_2 ) < ( v_2 - v_1 )

Therefore, \quad ( \tan \theta_{2} < \tan \theta_{1} ) .

- Slope of the graph is decreasing. Therefore, it is representing a retarded velocity motion of a body.
- Curve ( OB ) is a parabola which bends downwards.

See numerical problems based on this article.