Kinematics


   
 
Case of Uniform Acceleration
We know that
 
 
When a particle moves with uniform motion in a straight line in the given plane, the above equation always yields the same value of  , whatever may be the values of t and t'. However, when a particle moves in a plane with uniform acceleration, begins to depend on t and t'. This is because of change in velocity with time.
 
In the case of uniformly accelerated motion in a plane, we can still retain equation (1) but with a slight modification. Instead of velocity , we define another vector called average velocity vector. It is denoted by and is given by,
 
 
This equation refers to an 'interval of time' and not to an instant of time. However, there exists a velocity vector (t) at each instant of time just as there exists a position vector at each time t.
 
A particle is said to move with uniform acceleration if its velocity changes by equal amounts in equal intervals of time, however small these intervals may be.
 
Acceleration is the rate of the change in velocity to the corresponding time interval. It is denoted by  and is given by
 
As in the case of uniform velocity, the above equation will yield the same value of , whatever may be the values of t' and t. The manner of variation of velocity vector with time is given by the following equations.\
 
 
 
 
 
 
 
 
Next, consider how the position vector changes with time in the case of uniformly accelerated motion.
 
The average velocity in the time interval from t to is given by
 
 
 
 
 
 
The displacement undergone in the time interval from t to tl is given by
 
 
 
 
Substituting for Vav from equation (2) in the above equation we have,
 
 
 
This equation relates two different positions of a particle moving with uniform acceleration in a plane.
 
If we replace t' by t and t by 0,
 
 
In terms of rectangular components,
 
 
 
 
Each of these equations, when considered separately, represents uniformly accelerated motion in one dimension.
 

In equation (3),   are the position and velocity vectors respectively at t = 0. is the acceleration vector which is supposed to be constant.

 

In general, the vectors and  are not in the same direction.

 
Therefore, in order to find the sum of the three terms, the laws of vector addition will have to be used. The vector    t is in the same direction as itself but, its magnitude increases linearly with time. The vector is in the direction of . Its magnitude varies as the square of the time.
 
The figure below represents the following equation
 
 
 
Thus, we see that the velocity at any time t is the vector sum of the velocity at t = 0 and the additional velocity  acquired by the particle in time t due to uniform acceleration . Moreover, the same equation can also be expressed in the following two component forms, viz.
 
 
 
  
     
   
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Kinematics

  		•  Uniform Motion - Introduction
  		• Concept of a Point Object
  		• Rest and Motion
  		• Motion in One, Two and Three Dimensions
  		• Motion in a Straight Line
  		• Position-Time Graph for Non-Uniform Motion
  		• Graphical Representation of Uniform Motion
  		• Uniformly Accelerated Motion - Introduction
  		• Uniformly Accelerated Motion - Introduction
  		• Velocity-Time Graphs of Accelerated Motion
  		• Relations for Uniformly Accelerated Motion (Graphical Treatment)
  		• Position-Time Relations using Position-Time Graphs
  		• Distance as Area under Velocity-Time Graph
  		• Velocity-Time Relations using Calculus
  		• Velocity-Distance Relation
  		• Vectors - Introduction
  		• Definitions
  		• Unit Vector
  		• Fixed Vector
  		• Displacement Vector
  		• Multiplication of Vectors by Real Numbers, i.e., Scalar Multiple of a Vector
  		• Equality of Vectors
  		• The Zero Vector and its Properties
  		• Addition or Composition of Vectors
  		• Parallelogram Law of Vectors
  		• Law of Polygon of Vectors
  		• Resolution and Components of Vectors in a Plane
  		• Dot Product of Vectors and the Resolution of a Vector
  		• Relative Velocity in the Case of Two Velocities Inclined to Each Other
  		• Cross Product or Vector Product of Two Vectors
  		• Motion in a Plane - Introduction
  		• Case of Uniform Velocity
  		• Case of Uniform Acceleration
  		• Expression for Velocity Vector
  		• Projectile Motion
  		• Oblique Projectile
  		• Maximum Height
  		• Uniform Circular Motion
  		• Summary
    Animations
  		• Motion
  		• Difference between Distance and Displacement
  		• S - t Graph or Distance - Time Graph
  		• v - t Graph or Velocity - Time Graph
  		• Speed - Time Graph
  		• Scalar Product and Vector Product of two Vectors
  		• Projectile Motion