Acceleration is generally defined as the time rate of change of velocity. When can it be defined as the time rate of change of speed?

ANSWER: As we know that, acceleration is the rate of change of velocity. Mathematically, dV a = dt ----------------(1) In the above equation, it’s clear that acceleration is a vector quantity and is defined by how fast the velocity changes with respect to time. But in case of speed, it’s a scalar and has no specified direction. So, the acceleration can never be defined as the rate of change of speed. But, if the magnitude of velocity and speed are same and the direction is also same, at that very moment we can say that acceleration is the rate of change of speed. But still, its very clear that speed is a scalar and it’s differentiation w.r.t time will give another scalar. But acceleration is a vector quantity. So formally we cannot say that statement. If we consider the instantaneous velocity and speed, then the infinitesimal change in velocity and speed will be same. At this very small change the directions will not differ from each other a lot. So, these two can have the same meaning.