Blackboard showing different formulas. Image source: Pixabay.com

Derivation of the Acceleration

The notation

 

                      with

 

provides only the average acceleration between point 1 and point 2. The instantaneous acceleration a is the time derivative of the velocity vector

 

 

(bold characters are indicating a vector)

Equation: a=delta v/delta t
Equation: a=lim(delta T >0)(delta v / delta t)
The instantaneous acceleration a is the time derivative of the velocity

The same applies to the velocity. With r being the position vector The instantaneous velocity is

 

 

 

the acceleration becomes

Equation: v=lim(delta T >0)(delta r / delta t)
The instantaneous velocity is the time derivative of the position vector