What does the derivative of unit vector of velocity with respect to time represent?

Question

let an object move with a constant accelration a. in my book,the following derivatve is said to be non-constant(variable). $$\frac{d[\frac{v}{|v|}]}{dt}$$ what does this mean? as far as i can think,it should mean the rate of change of direction of velocity(as $\frac{v}{|v|}$ is a unit vector)if yes then what would its value be like

Bonus Question what would the graph of a body with negative accelration and initial positive velocity with respect to time look like?i think it should be a straight line on positive axis and then a sudden shift to a line on negative axis.

Answer 1

I think you don't find it strange that the derivative of the direction of the velocity vector is the acceleration perpendicular to the velocity vector. If the direction didn't change, this vector would be zero (contrary to the acceleration vector in the direction of motion of the particle).
If a particle has a positive velocity initially, its velocity is up the axis. If its velocity is zero, it's on the axis, and if its velocity is negative it's below the axis. The graph is indeed a straight line.