Question

What does $\left| {\dfrac{{dV}}{{dt}}} \right|$ and $\dfrac{{d\left| V \right|}}{{dt}}$ represents?

Answer 1

Hint:Acceleration is defined as the rate of change of velocity and it is the vector quantity which has magnitude as well as direction. Velocity is also a vector quantity. Whereas speed is a scalar quantity which has only magnitude and no direction. The speed is said to be constant when the object is moving in a circular path and velocity is not constant because the direction keeps on changing with respect to time continuously.

Complete step by step answer:
$ \Rightarrow \left| {\dfrac{{dV}}{{dt}}} \right|$
It represents rate of change of velocity (acceleration, because the equation represents that it is rate of change of velocity and we know that acceleration is defined as rate of change of velocity) and it is vector quantity which has both magnitude and direction.
$ \Rightarrow \dfrac{{d\left| V \right|}}{{dt}}$
It represents the rate of change of speed (magnitude of velocity vector) and it is a scalar quantity which has magnitude only and no direction. Common example for both the equation is an object moving at a constant rate in the circular path, $\left| {\dfrac{{dV}}{{dt}}} \right|$ Would point towards the centre because of centripetal force, centripetal force is required because to continue the motion in circular path. Whereas, $\dfrac{{d\left| V \right|}}{{dt}}$ Value will be equal to Zero, because the speed is constant.

Note:It can be said that rate of change of speed is same as that of rate of change of the magnitude of velocity\[\left\{ {\dfrac{{d\left| V \right|}}{{dt}}} \right\}\]. Therefore the rate of change of speed is not the same as that of rate of change of velocity. And also the rate of change of speed is equal in magnitude to the rate of change of velocity that is acceleration only when the object moves in a straight line without any reversal in direction.