A particle is moving in $x-y$ plane. At a certain instant of time, the components of its velocity and acceleration are as follows: $v_x=3 \mathrm{m} {\mathrm{s}}^{-1}, v_y=4 \mathrm{m} {\mathrm{s}}^{-1}, a_x=2 \mathrm{m} {\mathrm{s}}^{-2}$ and $a_y=1 \mathrm{m} {\mathrm{s}}^{-2}.$ Find the rate of change of speed (in $\mathrm{m} {\mathrm{s}}^{-2}$) at this moment.

A ball leaves a horizontal table with velocity $v_0=5 \mathrm{m} {\mathrm{s}}^{-1}.$ The ball bounces elastically from a vertical wall at a horizontal distance $D\left(=8 \mathrm{m}\right)$ from the table, as shown in figure. The ball then strikes the floor a distance $x_0$ from the table $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right).$ Find the value of $x_0$ (in $\mathrm{m}$).

Two particles are projected simultaneously from two points $O$ and $O\text{'}$ such that $10 \mathrm{m}$ is the horizontal and $5 \mathrm{m}$ is the vertical distance between them as shown in the figure. They are projected at the same inclination $60°$ to the horizontal with the same velocity $10 \mathrm{m} {\mathrm{s}}^{-1}$. Find the time (in sec) after which their separation becomes minimum.

Consider a disc rotating in the horizontal plane with a constant angular speed $\omega$ about its centre $O.$ The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles $P$ and $Q$ are simultaneously projected at an angle towards $R.$ The velocity of projection in the $y-z$ plane and is the same for both pebbles with respect to the disc. Assume that

$\left(\mathrm{i}\right)$ they land back on the disc before the disc has completed $\frac{1}{8}$ rotation,

$\left(\mathrm{ii}\right)$ their range is less than half the disc radius, and

$\left(\mathrm{iii}\right)$ $\omega$ remains constant throughout. Then