Question

An insect moves along a semi-circular transparent track of radius $R$, with a constant speed $V_0$. It starts from point O. A point source of light is fixed at the centre of the circle. What is the velocity of shadow of the insect on the surface at time $t$?

• A. $V_{0}sin\left(\frac{V_{0}t}{R}\right)$
• B. $V_{0}cose{c}^2\left(\frac{V_{0}t}{R}\right)$
• C. $V_{0}se{c}^2\left(\frac{R}{V_{0}t}\right)$
• D. $V_{0}se{c}^2\left(\frac{V_{0}t}{R}\right)$

Answer 1

Answer: (D)

$V_{0}se{c}^2\left(\frac{V_{0}t}{R}\right)$

In $t$ time, angular displacement of the insect:
$\theta =\frac{arc}{radius}=\frac{V_{0}t}{R}$
Distance of the shadow from point O is
$x=R\tan \theta$
$x=R\tan \left(\frac{V_{0}t}{R}\right)$
$\therefore$ Velocity $V=\frac{dx}{dt}=\left(R\right)\left(\frac{V_0}{R}\right){\sec }^2\left(\frac{V_{0}t}{R}\right)=V_0{\sec }^2\left(\frac{V_{0}t}{R}\right)$