Derivation of Equations of Motion under Uniform / Constant Velocity

1. The Vector Form of Equations of Motion under Uniform / Constant Velocity is derived as follows
   
   For any Object A moving with Uniform / Constant Velocity, the Velocity \(\vec{V_A}\) is calculated as the derivative of its Position Vector Function \(\vec{R_A(t)}\) with respect to time \(t\) as follows
   
   \({\Large \frac{d \vec{R_A(t)}}{dt}}=\vec{V_A}\)
   
   \(\Rightarrow d \vec{R_A(t)}=\vec{V_A}\hspace{1mm}dt\)   ...(1)
   
   Integrating equation (1) above gives the Displacement Vector Function \(\vec{R_{A_D}(t)}\) specifying Displacement of the Object A from the Initial Position after time \(t\) as follows
   
   \({\Large \int}d \vec{R_A(t)}={\Large \int}\vec{V_A}\hspace{1mm}dt\)
   
   \(\Rightarrow \vec{R_{A_D}(t)}=\vec{V_A}\hspace{1mm}t\)   ...(2)
   
   If the Inititial Constant Position Vector of the Object A is given by \(\vec{R_A(0)}\), the Position Vector Function of the Object A after time \(t\), \(\vec{R_A(t)}\), is given as
   
   \(\vec{R_A(t)}=\vec{R_A(0)} + \vec{R_{A_D}(t)}\)   ...(3)
   
   From equations (2) and (3) we get
   
   \(\vec{R_A(t)}=\vec{R_A(0)} + \vec{V_A}\hspace{1mm}t\)   ...(4)
   
   The equations (2) & (4) above give the Vector Form of Equations of Motion under Uniform / Constant Velocity where
   
   \(\vec{V_A}\) = Constant Velocity Vector with which the Object A is Moving
   
   \(\vec{R_A(0)}\) = Inititial Constant Position Vector of the Object A
   
   \(t\) = Ellapsed Time Interval
   
   \(\vec{R_{A_D}(t)}\) = Displacement Vector Function specifying Displacement of the Object A from the Initial Position after time \(t\)
   
   \(\vec{R_A(t)}\) = Position Vector Function of the Object A after time \(t\)
   
   The equation (4) can be used to Find Trajectory / Path of the Object under Uniform / Constant Velocity. Please note that this equation is similar to the Position Vector Equation of a Line. Hence, it can be said that Any Object under Uniform / Constant Velocity has a Linear Trajectory.
2. The Scalar Form of Equation of Motion under Uniform / Constant Velocity for any Object A is given by equations (5) as follows
   
   \(D_A(t)= S_At\)   ...(5)
   
   where
   
   \(S_A\) = Constant Speed with which the Object A is Moving
   
   \(t\) = Ellapsed Time Interval
   
   \(D_A(t)\) = Distance Travelled by the Object A after time \(t\)
   
   Please note that the Speed of Object A \(S_A\) can be calculated as the Magnitude of its Velocity Vector \(\vec{V_A}\). Similarly, the Distance Travelled by Object A \(D_A(t)\) can also be calculated as the Magnitude of its Displacement Vector \(\vec{R_{A_D}(t)}\). That is,
   
   \(S_A=|\vec{V_A}|\)   ...(4)
   
   \(D_A(t)=|\vec{R_{A_D}}(t)|\)   ...(5)