Question

A particle is moving in a plane with velocity given by $y={\nu }_{0}i+\left(\alpha \omega cos\omega t\right)j,$where i , j are units vectors along $x$ and $y$axes respectively .The trajectory of the particle if the particle starts from origin at $t=0$ will be

• A. $y=\alpha sin\frac{\omega x}{{\nu }_0}$
• B. $y=\alpha cos\frac{\omega x}{{\nu }_0}$
• C. $y=tanx$
• D. $y=cotx$

Answer 1

Answer: (A)

$y=\alpha sin\frac{\omega x}{{\nu }_0}$

Velocity , $V={\nu }_{0}i+\alpha \omega cos\omega t$
X component , ${\nu }_x={\nu }_0$
$\Rightarrow$ $\frac{dx}{dt}={\nu }_0$
$\Rightarrow$ $dx={\nu }_{0}dt$
$\Rightarrow$ $x={\nu }_{0}t$
$\Rightarrow$ $t=\frac{x}{{\nu }_0}$...............(i)
Y component , ${\nu }_y=\alpha \omega cos\omega t$
$\Rightarrow$ $\frac{dy}{dt}=\alpha \omega cos\omega t$
$\Rightarrow$ $dy=\alpha \omega cos\omega tdt$
$\Rightarrow$ $y=\frac{\alpha \omega sin\omega t}{\omega }$
$\Rightarrow$ $\alpha sincot$
From Eq.(i) ,
$y=\alpha sin\omega \frac{x}{{\nu }_0}$