# A particle moves in xy plane. The position vector of particle at any time t is $\overrightarrow r=\{(2t) i+(2t^2)j\}\; m$. The rate of change of $\theta$ at time $t= 2\;$ seconds is ( $\theta$ is the angle which its velocity vector makes with positive x-axis )

$(a)\;\frac{2}{17}rad/s\quad (b)\;\frac{1}{14}rad/s \quad (c)\;\frac{4}{7}rad/s \quad (d)\;\frac{6}{5}rad/s$
Can you answer this question?

$x=2t$
=>$v_x=\large\frac{dx}{dt}$$=2 y=2t^2 =>v_y=\large\frac{dy}{dt}$$=4t$
$\tan \theta=\large\frac{v_y}{v_x}$
$\qquad=\large\frac{4t}{2}$$=2t Differentiating with respect to t (\sec^2 \theta) \large\frac{d\theta}{dt}$$=2$
$(1+\tan ^2 \theta) \large\frac{d\theta}{dt}$$=2 (1+4t^2) \large\frac{d\theta}{dt}$$=2$
$\large\frac{d\theta}{dt}=\large\frac{2}{1+4t^2}$
at $t=2$
$\large\frac{d\theta}{dt}=\frac{2}{1+4(2)^2}=\frac{2}{17}$$rad/s$
Hence a is the correct answer.

answered Jun 29, 2013 by
edited Jan 25, 2014 by meena.p

+1 vote

+1 vote