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Graphical Representation of Linear Motion

When a body moves along a straight line path, its motion is said to be linear or rectilinear. The linear motion of a body can be studied with the help of the following graphs.
  1. Displacement–time graph
  2. Velocity–time graph
  3. Acceleration–time graph

Displacement–Time Graph

A graph showing the position of a moving object with time is called the displacement–time graph.

In the displacement–time graph, the time is plotted on the X-axis and the displacement of the body is plotted on the Y-axis (Figure 2.2). We have,
 
Description: 45756.png

Therefore, the slope of the displacement–time graph gives the velocity.
 


 

Slope of the line l is m = tan θ = Description: 51729.png
 
where θ is the angle between the straight line and the positive X-axis.
 

  1. The displacement–time graph for a stationary body is a straight line parallel to the time axis as shown in Figure 2.3.

Since the slope is zero, the velocity is zero.
  1. Displacement–time graph of a body with uniform velocity
     
    The displacement–time graph is a straight line inclined to the time axis for a body moving with uniform velocity as shown in Figure 2.4.
Time in second (s)
0
1
2
3
4
5
Displacement in metre (m)
0
10
20
30
40
50

 

 

From the graph, we observe that for a body moving with uniform velocity, the displacement is proportional to the time (i.e., s  t).

The velocity of the body can be obtained by finding the slope of the straight line OP. Therefore,
 
Velocity of the body = Slope of the line OP
 
Description: 46012.png
  1. Displacement–time graph of a body with non-uniform velocity
     
    If a body moves with uniform acceleration, the displacement–time graph is a curve as shown in Figure 2.5.


 

The velocity at any instant can be obtained by finding the slope of the tangent drawn to the curve at that instant.

In Figure 2.5, velocity at t = 5 s or when the displacement s = 30 m is obtained by finding the slope of the tangent to the curve drawn at A corresponding to t = 5 or s = 30 m.
 
Description: 46085.png
 
If the slope of the displacement–time graph is negative, the body is returning towards its starting point (reference point) or moving in the negative direction (−s).
 

Velocity–Time Graph

The graph that shows the variation in velocity of a moving object with the passage of time is called the velocity–time graph.

In the velocity–time graph, the time is plotted on the X-axis and the velocity is plotted on the Y-axis.

Since, velocity  time = displacement, the area enclosed between the velocity–time graph and X-axis (i.e., time axis) gives the displacement of a body. In addition, since acceleration is the rate of change in velocity with time, the slope of v–t graph gives the acceleration of the body.


Case (i)
 
The velocity–time graph is a straight line parallel to the time axis for a body moving with uniform velocity as shown in Figure 2.6.
 

 

  • In Figure 2.6, the slope of the straight line AB is zero. Therefore, its acceleration is zero.
  • The displacement in 6 s = area of rectangle OABC OC × BC = 6 × 4 = 24 m.


Case (ii)
 
If a body is in uniform acceleration (i.e., equal changes in velocity take place in equal intervals of time), the velocity–time graph is a straight line inclined to the time axis. The slope of the line gives the acceleration.
 


 

In Figure 2.7, the distance travelled by the body in 5 s = Area of triangle ∆OMN.
Distance travelled Description: 46313.png
 
Acceleration of the body = Slope of the line ON
 
Description: 46326.png
 

Case (iii)
 
If the body is in motion with variable acceleration, the velocity–time graph is a curve (Figure 2.8).
 


 

Total displacement in 7 s = Area of shaded portion (approximately area of 12 squares).
 
The squares with less than half the portion within the curve are ignored.

Acceleration–Time Graph

The graph that shows the variation in acceleration of a moving object with the passage of time is called the acceleration–time graph.
 
In the acceleration–time graph, the time is plotted on the X-axis and the acceleration is plotted on the Y-axis.
 
Since, acceleration  time = change in velocity, the area enclosed between the acceleration–time graph and the time axis gives the change in the velocity of the body.
 
Case (i) If the body is stationary or it is moving with a uniform velocity, then the acceleration is zero. The acceleration–time graph in such a case is a straight line coinciding with the time axis.
 
Case (ii) If the velocity of the body in motion increases uniformly with time, the acceleration is constant. Then, the acceleration–time graph is a straight line parallel to the time axis as shown in Figure 2.9.
 
Case (iii) If the velocity of the body decreases at a constant rate, the retardation is constant. The acceleration–time graph is a straight line parallel to the time axis on the negative acceleration axis as shown in Figure 2.10.
 

 





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