Velocity and Speed of an Object in Motion and Absolute / Relative Velocities

1. Velocity (also called Instantaneous Velocity) of any Object is the Rate at which the Position of the Object changes. Velocity for any Object A at any instance of time \(t\), denoted by \(\vec{V_A(t)}\), is given by the Derivative of its Position Vector Function \(\vec{R_A(t)}\) with respect to time variable \(t\) as follows
   
   \(\vec{V_{A}(t)}={\Large \frac{d \vec{R_{A}(t)}}{dt}}\)   ...(1)
   
   Please note that it is same as Finding Tangent of a Curve given its Parametric Equation.
2. Speed (also called Instantaneous Speed) of any Object at any instance of time is the Magnitude of its Velocity Vector and it is a Scalar Quantity. Speed for any Object A at any instance of time \(t\), denoted by \(S_{A}(t)\), is given as follows
   
   \(S_{A}(t) = |\vec{V_{A}(t)}|\)   ...(2)
   
   The Speed of the Object A as given in equation (2) is a Scalar Function of time Variable \(t\). This function can be used for finding the Distance Travelled by the Object A over a Path / Trajectory in a given time interval.
3. As given above, the Absolute Velocity of Any Object is Rate at which the Position of the Object changes. Given any 2 Objects \(A\) and \(B\), their Absolute Velocities at an instance of time \(t\) are given as \(\vec{V_A(t)}\) and \(\vec{V_B(t)}\) respectively.
4. The Relative Velocity of Any Object \(A\) with respect to Any Other Object \(B\) at an instance of time \(t\) (\(\vec{V_{AB}(t)}\)) is given by the Difference of their Absolute Velocity Vectors as follows
   
   \(\vec{V_{AB}(t)}=\vec{V_A}(t)-\vec{V_B(t)}\)   ...(3)
   
   Similarly, the Relative Velocity of the Object \(B\) with respect to Object \(A\) at an instance of time \(t\) (\(\vec{V_{BA}}\)) is given by the Difference of their Absolute Velocity Vectors as follows
   
   \(\vec{V_{BA}(t)}=\vec{V_B(t)}-\vec{V_A(t)}\)   ...(4)
   
   Therefore,
   
   \(\vec{V_{AB}(t)} = -\vec{V_{BA}(t)}\)   ...(5)
   
   
5. If Any 2 Sets of Relative Velocities between 3 Objects are known, then the Third Set can be found out by Triangle Law of Vector Addition/Subtraction. For example, given any 3 Objects \(A\), \(B\) and \(C\) having Absolute Velocities \(\vec{V_A}(t)\), \(\vec{V_B(t)}\) and \(\vec{V_C(t)}\) respectively, then
   
   \(\vec{V_{AC}(t)}=\vec{V_{AB}(t)} + \vec{V_{BC}(t)}\)   ...(6)   (as per Triangle Law of Vector Addition/Subtraction)
   
   Using the above formula, if any 2 of \(\vec{V_{AB}(t)}\) (or \(\vec{V_{BA}(t)}\)) , \(\vec{V_{BC}(t)}\) (or \(\vec{V_{CB}(t)}\)) and \(\vec{V_{AC}(t)}\) (or \(\vec{V_{CA}(t)}\)) are known, the third can be found out.
6. Please note that the formulae for Relative Velocities given in equations (3), (4), (5) & (6) are applicable only when \(\vec{V_A(t)}\), \(\vec{V_B(t)}\) and \(\vec{V_C(t)}\) are all Very Small as Compared to Velocity of Light in Vacuum.
   
   When the Velocity of Any One of the Objects is comparable to Velocity of Light in Vacuum, Einstein Velocity Addition/Subtraction Formula for Special Relativity is used for finding Relative Velocities. Under such circumstances \(\vec{V_{AB}(t)}\neq-\vec{V_{BA}(t)}\) and hence the Triangle Law of Vector Addition/Subtraction for Relative Velocities is Not Applicable.