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Calculation of distance and displacement

The area covered between the velocity–time graph and time axis gives the displacement and distance traveled by the body for a given time interval.
 
Total distance = | A1 | + | A2 | + | A3 |
= Addition of modulus of different area,
i.e., 24696.png
 
Total displacement = A1 + A2 + A3
= Addition of different area considering their sign
i.e., 24709.png
 
The area above time axis is taken as positive, while the area below time axis is taken as negative.
 
26640.png
 
Here, A1 and A2 are the area of triangles 1 and 2, respectively, and A3 is the area of trapezium (Figure).

Calculation of acceleration

The slope of tangent on velocity–time graph represents the acceleration of the particle.
 
During the motion from O  A, acceleration is positive.
 
During the motion from A  B, acceleration is negative.
 
During the motion from B  C, acceleration is positive.
 
During the motion from C  D, acceleration is zero.
 
During the motion from D  E, acceleration is negative.

Graph

  1. 26718.png
Slope = 0, acceleration = 0, v = constant, i.e., line parallel to time axis represents that the particle is moving with constant velocity.
 
 
 
  1. 26728.png
Slope is infinite hence acceleration is infinite, i.e., line perpendicular to time axis represents that the particle is increasing its velocity, but time does not change. It means the particle possesses infinite acceleration.
 
Practically, it is not possible.
 
 
 
  1. 26738.png
Slope is constant and positive hence acceleration is constant and v is increasing uniformly with time, i.e., line with constant slope represents uniform acceleration of the particle.
 
 
 
  1. 26748.png
Slope is increasing, the acceleration is not constant. Acceleration is increasing.
 
 
 
  1. 26758.png
Slope is decreasing, the acceleration is not constant. Acceleration is decreasing.
 
 
 
  1. 26768.png
Slope is positive but constant. Hence, acceleration will be positive and constant.
 
 
 
  1. 26781.png
Slope is negative but constant. Hence, acceleration will be negative and constant.

Some Important Notes

  • For two particles having displacement time graph with slopes θ1 and θ2 and possessing velocities v1 and v2 , respectively, 29537.png.
  • Greater the slope of displacement-time graph, greater is the velocity and vice-versa.
  • Area under v–t graph = displacement of the particle.




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