Question

Pick the correct statements:
(a) Average speed of a particle in a given time is never less than the magnitude of the average velocity.
(b) It is possible to have a situation in which $\left|\frac{d\overrightarrow{v}}{dt}\right|\ne 0\mathrm{but}\frac{d}{dt}=\left|\overrightarrow{v}\right|=0$
(c) The average velocity of a particle is zero in a time interval. It is possible that the instantaneous velocity is never zero is the interval.
(d) The average velocity of a particle moving on a straight line is zero in a time interval. It is possible that the instantaneous velocity is never zero in the interval. (Infinite accelerations are not allowed).

Answer 1

(a) Average speed of a particle in a given time is never less than the magnitude of the average velocity.
(b) It is possible to have a situation in which $\left|\frac{d\overrightarrow{v}}{dt}\right|\ne 0\mathrm{but}\frac{d}{dt}\left|\overrightarrow{v}\right|=0$.
(c) 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, for example, the motion of a particle on a circular track with a constant speed.
Average velocity = $\frac{\mathrm{Displacement}}{\mathrm{Total}\mathrm{time}}$
Displacement ≤ Distance
$\therefore$ Average velocity ≤ Average speed
In uniform circular motion, speed is constant but velocity is not.
i.e., $\left|\frac{d\overrightarrow{v}}{dt}\right|\ne 0\mathrm{but}\frac{d}{dt}=\left|\overrightarrow{v}\right|=0$ which proves case (b)

(d) In one complete circle of uniform motion, average velocity is zero. Instantaneous velocity is never zero in the interval.