Derivation of Equations of Motion under Uniform / Constant Acceleration

1. The Vector Form of Equations of Motion under Uniform / Constant Acceleration is derived as follows
   
   For any Object A moving with Uniform / Constant Acceleration, the Acceleration \(\vec{A_A}\) is calculated as the derivative of its Velocity Vector Function \(\vec{V_A(t)}\) with respect to time \(t\) as follows
   
   \({\Large \frac{d \vec{V_A(t)}}{dt}}=\vec{A_A}\)
   
   \(\Rightarrow d \vec{V_A(t)}=\vec{A_A}\hspace{1mm}dt\)   ...(1)
   
   Integrating equation (1) above gives the Velocity Displacement Vector Function \(\vec{V_{A_D}(t)}\) specifying Difference between Velocity of the Object A at time t and the Initial Velocity as follows
   
   \({\Large \int}d \vec{V_A(t)}={\Large \int}\vec{A_A}\hspace{1mm}dt\)
   
   \(\Rightarrow \vec{V_{A_D}(t)}=\vec{A_A}\hspace{1mm}t\)   ...(2)
   
   If the Inititial Constant Velocity Vector of the Object A is given by \(\vec{V_A(0)}\), the Velocity Vector Function of the Object A after time \(t\), \(\vec{V_A(t)}\), is given as
   
   \(\vec{V_A(t)}=\vec{V_A(0)} + \vec{V_{A_D}(t)}\)   ...(3)
   
   From equations (2) and (3) we get
   
   \(\vec{V_A(t)}=\vec{V_A(0)} + \vec{A_A}\hspace{1mm}t\)   ...(4)
   
   Now, the Velocity of Object A, \(\vec{V_A(t)}\), can also be calculated as the derivative of its Position Vector Function \(\vec{R_A(t)}\) with respect to time \(t\) as follows
   
   \({\Large \frac{d \vec{R_A(t)}}{dt}}=\vec{V_A(t)}\)   ...(5)
   
   From equations (4) and (5) we get
   
   \({\Large \frac{d \vec{R_A(t)}}{dt}}=\vec{V_A(0)} + \vec{A_A}\hspace{1mm}t\)
   
   \(\Rightarrow d \vec{R_A(t)}=(\vec{V_A(0)} + \vec{A_A}\hspace{1mm}t)\hspace{1mm}dt\)   ...(6)
   
   Integrating equation (6) 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
   
   \({\Large \int}d \vec{R_A(t)}={\Large \int}(\vec{V_A(0)} + \vec{A_A}\hspace{1mm}t)\hspace{1mm}dt\)
   
   \(\Rightarrow \vec{R_{A_D}(t)}=\vec{V_A(0)}\hspace{1mm}t + {\Large \frac{1}{2}}\vec{A_A}\hspace{1mm}t^2\)   ...(7)
   
   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
   
   \(\vec{R_A(t)}=\vec{R_A(0)} + \vec{R_{A_D}(t)}\)   ...(8)
   
   From equations (7) and (8) we get
   
   \(\vec{R_A(t)}=\vec{R_A(0)} + \vec{V_A(0)}\hspace{1mm}t + {\Large \frac{1}{2}}\vec{A_A}\hspace{1mm}t^2\)   ...(9)
   
   Now, in equations (6) we have
   
   \(\vec{V_A(t)}=\vec{V_A(0)} + \vec{A_A}\hspace{1mm}t\)
   
   \(\vec{V_A(t)}-\vec{V_A(0)}=\vec{A_A}\hspace{1mm}t\)   ...(10)
   
   \(\Rightarrow \vec{A_A}={\Large \frac{\vec{V_A(t)}-\vec{V_A(0)}}{t}}\)   ...(11)
   
   Now, putting the value of Constant Acceleration \(\vec{A_A}\) from equation (11) above in equations (9) we get
   
   \(\vec{R_{A_D}(t)}=\vec{V_A(0)}\hspace{1mm}t + {\Large \frac{1}{2}}({\Large \frac{\vec{V_A(t)}-\vec{V_A(0)}}{t}})\hspace{1mm}t^2\)
   
   \(\Rightarrow \vec{R_{A_D}(t)}={\Large (\frac{\vec{V_A(t)}+\vec{V_A(0)}}{2})}\hspace{1mm}t\)   ...(12)
   
   \(\Rightarrow 2\vec{R_{A_D}(t)}=(\vec{V_A(t)}+\vec{V_A(0)})\hspace{1mm}t\)   ...(13)
   
   Now, taking Dot Product of Constant Acceleration \(\vec{A_A}\) with equation (13) we get
   
   \(2(\vec{A_A} \cdot \vec{R_{A_D}(t)})= t\hspace{1mm}\vec{A_A} \cdot (\vec{V_A(t)}+\vec{V_A(0)})\)   ...(14)
   
   Now, putting the value of \(\vec{A_A}\hspace{1mm}t\) from equation (10) above in equations (14) we get
   
   \(2(\vec{A_A} \cdot \vec{R_{A_D}(t)})= (\vec{V_A(t)}-\vec{V_A(0)}) \cdot (\vec{V_A(t)}+\vec{V_A(0)})\)
   
   \(\Rightarrow 2(\vec{A_A} \cdot \vec{R_{A_D}(t)})= \vec{V_A(t)}\cdot\vec{V_A(t)}-\vec{V_A(0)}\cdot\vec{V_A(0)}\)
   
   \(\Rightarrow 2(\vec{A_A} \cdot \vec{R_{A_D}(t)})= {V_A(t)}^2-{V_A(0)}^2\)   ...(15)
   
   The equations (2), (4), (7), (9), (12) and (15) above give the Vector Form of Equations of Motion under Uniform / Constant Acceleration. The same equations have been summarized below
   1. \(\vec{V_{A_D}(t)}=\vec{A_A}\hspace{1mm}t\)   ...(2)
   2. \(\vec{V_A(t)}=\vec{V_A(0)} + \vec{A_A}\hspace{1mm}t\)   ...(4)
   3. \(\vec{R_{A_D}(t)}=\vec{V_A(0)}\hspace{1mm}t + {\Large \frac{1}{2}}\vec{A_A}\hspace{1mm}t^2\)   ...(7)
   4. \(\vec{R_A(t)}=\vec{R_A(0)} + \vec{V_A(0)}\hspace{1mm}t + {\Large \frac{1}{2}}\vec{A_A}\hspace{1mm}t^2\)   ...(9)
   5. \(\vec{R_{A_D}(t)}={\Large (\frac{\vec{V_A(t)}+\vec{V_A(0)}}{2})}\hspace{1mm}t\)   ...(12)
   6. \(2(\vec{A_A} \cdot \vec{R_{A_D}(t)})= {|\vec{V_A(t)}|}^2-{|\vec{V_A(0)}|}^2\)   ...(15)
   where
   
   \(\vec{A_A}\) = Constant Acceleration Vector of Object A
   
   \(\vec{V_A(0)}\) = Initial Constant Velocity Vector of Object A
   
   \(\vec{R_A(0)}\) = Inititial Constant Position Vector of the Object A
   
   \(t\) = Ellapsed Time Interval
   
   \(\vec{V_{A_D}(t)}\) = Velocity Displacement Vector Function specifying Difference between Velocity of the Object A at time t and the Initial Velocity
   
   \(\vec{V_A(t)}\) = Velocity Vector Function of the Object A after time \(t\)
   
   \(\vec{R_{A_D}(t)}\) = 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 (9) can be used to Find Trajectory / Path of the Object under Uniform / Constant Acceleration. Please note that this equation is similar to
   1. Position Vector Equation of a Line when the Initial Velocity Vector of the Object \(\vec{V_A(0)}\) is in Same or Opposite Direction of Acceleration Vector \(\vec{A_A}\). Hence, it can be said that under such cases Any Object under Uniform / Constant Acceleration has a Linear Trajectory.
   2. Position Vector Equation of a Parabola when the Initial Velocity Vector of the Object \(\vec{V_A(0)}\) is Not in Same or Opposite Direction of Acceleration Vector \(\vec{A_A}\)). Hence, it can be said that under such cases Any Object under Uniform / Constant Acceleration has a Parabolic Trajectory.
2. The Scalar Form of Equation of Motion under Uniform / Constant Acceleration for any Object A is given as follows
   1. \(S_{A_D}(t)= A_A\hspace{1mm}t\)   ...(16)
   2. \(S_A(t)= S_A(0) + S_{A_D}(t) \Rightarrow S_A(t)= S_A(0) + A_A\hspace{1mm}t\)   ...(17)
   3. \(D_A(t)= S_A(0)\hspace{1mm}t + {\Large \frac{1}{2}}A_A\hspace{1mm}t^2\)   ...(18)
   4. \(D_A(t)= {\Large (\frac{S_A(t)+ S_A(0)}{2})}\hspace{1mm}t\)   ...(19)
   5. \(2 A_A D_A(t)= {S_A(t)}^2-{S_A(0)}^2\)   ...(20)
   where
   
   \(A_A\) = Magnitude of the Constant Acceleration Vector of Object A
   
   \(S_A(0)\) = Initial Constant Speed or Magnitude of the Initial Constant Velocity Vector of Object A
   
   \(t\) = Ellapsed Time Interval
   
   \(S_{A_D}(t)\) = Change in Speed of Object A in time t
   
   \(S_A(t)\)= Speed of Object A after time t
   
   \(D_A(t)\) = Distance Travelled by the Object A after time \(t\)
   
   Please note that Scalar Form of Equation of Motion under Uniform / Constant Acceleration are valid only when Initial Velocity Vector of the Object \(\vec{V_A(0)}\) is in Same Direction of Acceleration Vector \(\vec{A_A}\) (i.e when the Object A has a Linear Trajectory). Only under such condition \(S_{A_D}(t)=|\vec{V_{A_D}(t)}|\) and \(D_A(t)=|\vec{R_{A_D}(t)}|\).
   
   If the Initial Velocity Vector of the Object \(\vec{V_A(0)}\) is in Different Direction than Acceleration Vector \(\vec{A_A}\) (i.e when the Object A has a Parabolic Trajectory), the calculations obtained through Scalar Form of Equation of Motion under Uniform / Constant Acceleration do not correspond to the calculations obtained through Vector Form of Equation of Motion under Uniform / Constant Acceleration. Under such condition \(S_{A_D}(t)\neq|\vec{V_{A_D}(t)}|\) and \(D_A(t)\neq|\vec{R_{A_D}(t)}|\)