- Average speed of a particle in a given time is never less than the magnitude of the average velocity.
- It is possible to have a situation in which $\Big|\frac{\text{d}\vec{\text{v}}}{\text{dt}}\Big|\neq0$ but $\frac{\text{d}}{\text{dt}}|\vec{\text{v}}|=0.$
- The average velocity of a particle is zero in a time interval. It is possible that the instantaneous velocity is never zero in the interval.
Explanation:
- Average speed of a particle in a given time is never less than the magnitude of the average velocity.
- It is possible to have a situation in which $\Big|\frac{\text{d}\vec{\text{v}}}{\text{dt}}\Big|\neq0$ but $\frac{\text{d}}{\text{dt}}|\vec{\text{v}}|=0.$
- The average velocity of a particle is zero in a time interval. It is possible that the instantaneous velocity is never zero in the interval.
Example, the motion of a particle on a circular track with a constant speed.
Average velocity $=\frac{\text{Displacement}}{\text{Total time}}$
$\text{Displacement}\leq\text{Distance}$
$\therefore\text{Average velocrty}\leq\text{Average speed}$
In uniform circular motion, speed is constant but velocity is not.
$\text{i.e.},\Big|\frac{\text{d}\vec{\text{v}}}{\text{dt}}\Big|\neq0$ but $\frac{\text{d}}{\text{dt}}=|\vec{\text{v}}|=0$ which proves case (b)
- In one complete circle of uniform motion, average velocity is zero. Instantaneous velocity is never zero in the interval.
