Question

The velocity of a particle moving in a straight line varies with time in such a manner that $v$ versus $t$ graph is represented by one-half of an ellipse. The maximum velocity is $v_m$ and the total time of motion is $t_0$.

(i) average velocity of the particle is $\frac{\pi}{4} v_m$
(ii) such motion cannot be realized in practical terms

• A. only (i) is correct
• B. only (ii) is correct
• C. both (ii) and (ii) are correct
• D. both (i) and (ii) are wrong

Answer 1

Answer: (C)

The displacement of the particle is determined by the area bounded by the curve. This area is:
$s = \frac{1}{2}\pi v_m \frac{t_0}{2}$
The average velocity is: $< v > = \frac{s}{t_0} = \frac{\pi}{4}v_m$
Such motion cannot be realized in practical terms since at the initial and final moments, the acceleration (which is slope of $v - t$ graph) is infinitely large.