Question

If $\overrightarrow{u}$ is instantaneous velocity of particle and $\overrightarrow{v}$ is velocity of wave , then

• A. $\overrightarrow{u}$ is perpendicular $\overrightarrow{v}$
• B. $\overrightarrow{u}$ is parallel to $\overrightarrow{v}$
• C. $\left|\overrightarrow{u}\right|$ is equal to $\left|\overrightarrow{v}\right|$
• D. $\left|\overrightarrow{u}\right|$ is equal to (slope of wave form )$\left|\overrightarrow{v}\right|$

Answer 1

The correct option is D $\left|\overrightarrow{u}\right|$ is equal to (slope of wave form )$\left|\overrightarrow{v}\right|$

We know that equation of a wave with velocity v.

$y=asin\omega (t-\frac{x}{v})$...(i)

differentiating (i) with respect to time 't' at some instant $x=x_0$

$\frac{dy}{dt}=Awcosw(t-\frac{x_0}{v})$

differentiate (i) with respect to x

$\frac{dy}{dx}=-\frac{Aw}{v}cosw(t-\frac{x}{v})$

${\left(\frac{dy}{dx}\right)}_{x_0}=-\frac{Aw}{v}cosw(t-\frac{x_0}{v})$...(iii)

from (ii)

${\left(\frac{dy}{dx}\right)}_{x_0}=-\frac{1}{v}\left(\frac{dy}{dx}\right)$

$\Rightarrow \frac{dy}{dx}=-{\left(\frac{dy}{dx}\right)}_{x_0}v$

$\Rightarrow |\overrightarrow{u}|=$ (slope of wave from) $|\overrightarrow{v}|$