$P,Q$ and $R$ are three particles of a medium which lie on the $x$-axis. A sine wave of wavelength $\lambda$ is travelling through the medium in the $x$-direction. $P$ and $Q$ always have the same speed, while $P$ and $R$ always have the same velocity. The minimum distance between:-

$y_1=16\sin \omega t \mathrm{mm}, y_2=3\sin \left(\omega t-\frac{\pi }{2}\right)\mathrm{mm}, y_3=7\sin \left(\omega t+\frac{\pi }{2}\right)\mathrm{mm}$ & $y_4=-13\sin \omega t\mathrm{mm}$

In the given figure the string has mass $4.5 \mathrm{g}$. If the time taken by a transverse pulse produced at the floor to reach the pulley is $10\mathrm{p}$ ms then $\mathrm{p}$ is : $\left(g=10{ \mathrm{m} \mathrm{s}}^{-2}\right)$.

A string vibrate according to the equation $y=5\sin \left(\frac{\pi x}{3}\right)\cos \left(40\pi t\right)$ where $x$ and $y$ are in $\mathrm{cm}\text{'}s$ and $t$ is in second.