Position, Displacement and Trajectory of an Object in Motion

1. Position of any Object in Motion is given by its Position Vector Function.
2. The Position Vector Function of any Object is a Position Vector whose Components are Functions of a Single Variable Parameter. This Parameter is Time Interval During which the Motion has taken place and is denoted by \(t\). The Actual Position Vector for the Object at any instance of time is calculated by Providing a Value to this Variable Parameter \(t\).
   
   For e.g, for any Object \(A\) which is in Motion, its Position Vector Function is denoted by \(\vec{R_A(t)}\).
   
   Initial Position of Object \(A\) at time \(t=0\) is given by Position Vector \(\vec{R_A(0)}\)
   
   Position of Object \(A\) at time \(t=t_n\) is given by Position Vector \(\vec{R_A(t_n)}\)
   
   \(\therefore\), Displacement of Object \(A\) from Initial Position at time \(t=t_n\) is given by the Displacement Vector \(\vec{R_{A_D}(t_n)}\) = \(\vec{R_A(t_n)} - \vec{R_A(0)}\)
   
   \(\Rightarrow\) Position of Object \(A\) at time (\(t=t_n\)) = \(\vec{R_A(t_n)}\) = \(\vec{R_A(0)} + \vec{R_{A_D}(t_n)}\)
3. The Trajectory of an Object is the Path Traced by the Object while in Motion which is given by the Curve Traced by the Coordinate Points given by Position Vectors over a given interval of time. For e.g, for any Object \(A\) in Motion having Position Vector Function \(\vec{R_A(t)}\), its Position Vectors in the time interval from \(t=0\) to \(t=t_n\) can be given as
   
   Initial Position at time (\(t=0\)) = \(\vec{R_A(0)}\)
   
   Position at time (\(t=t_1\)) = \(\vec{R_A(t_1)} = \vec{R_A(0)} + \vec{R_{A_D}(t_1)}\)
   
   Position at time (\(t=t_2\)) = \(\vec{R_A(t_2)} = \vec{R_A(0)} + \vec{R_{A_D}(t_2)}\)
   
   Position at time (\(t=t_3\)) = \(\vec{R_A(t_3)} = \vec{R_A(0)} + \vec{R_{A_D}(t_3)}\)
   
   \(\vdots\)
   
   Position at time (\(t=t_n\)) = \(\vec{R_A(t_n)} = \vec{R_A(0)} + \vec{R_{A_D}(t_n)}\)
   
   Hence, its Trajectory from time \(t=0\) to \(t=t_n\) is given by the Curve Traced by the Coordinate Points given by the Position Vectors \(\vec{R_A(0)},\hspace{1mm}\vec{R_A(t_1)},\hspace{1mm}\vec{R_A(t_2)},\hspace{1mm}\vec{R_A(t_3)},\hspace{1mm}\cdots,\hspace{1mm}\vec{R_A(t_n)}\).