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

\[\left| \frac{d \vec{v}}{dt} \right| \neq 0 \text{ but } \frac{d}{dt} = \left| \vec{v} \right| = 0\]

(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 \[\left| \frac{d \vec{v}}{dt} \right| \neq 0 \text{ but } \frac{d}{dt}\left| \vec{v} \right| = 0\] .

(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 = \[\frac{\text{ Displacement } }{\text{ Total time} }\]

Displacement ≤ Distance

∴ Average velocity ≤ Average speed  In uniform circular motion, speed is constant but velocity is not.
i.e., \[\left| \frac{d \vec{v}}{dt} \right| \neq 0 \text{ but } \frac{d}{dt} = \left| \vec{v} \right| = 0\]  which proves case (b) 

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