A body of mass $m$ is launched up on a rough inclined plane making an angle of $30°$ with the horizontal. The coefficient of friction between the body and plane is $\frac{\sqrt{x}}{5}$, if the time of ascent is half of the time of descent. The value of $x$ is

Two billiard balls of equal mass $30 \mathrm{g}$ strike a rigid wall with same speed of $108 \mathrm{km} {\mathrm{h}}^{-1}$ (as shown) but at different angles. If the balls get reflected with the same speed, then the ratio of the magnitude of impulses imparted to ball $a$ and ball $b$ by the wall along $X$ direction is:

A block of mass $m$ slides along a floor while a force of magnitude $F$ is applied to it at an angle $\theta$ as shown in figure. The coefficient of kinetic friction is ${\mu }_K$. Then, the block's acceleration $a$ is given by : ($g$ is acceleration due to gravity)

A ball of mass $10 \mathrm{kg}$ moving with a velocity $10\sqrt{3}{ \mathrm{m} \mathrm{s}}^{-1}$ along $X$-axis, hits another ball of mass $20 \mathrm{kg}$ which is at rest. After the collision, the first ball comes to rest and the second one disintegrates into two equal pieces. One of the pieces starts moving along $Y$-axis at a speed of $10 \mathrm{m} {\mathrm{s}}^{-1}$. The second piece starts moving at a speed of $20 \mathrm{m} {\mathrm{s}}^{-1}$ at an angle $\theta$ (degree) with respect to the $X$-axis. The configuration of pieces after the collision is shown in the figure. The value of $\theta$(in degrees) to the nearest integer is _________.

Statement I: A cyclist is moving on an unbanked road with a speed of $7 \mathrm{km} {\mathrm{h}}^{-1}$ and takes a sharp circular turn along a path of the radius of $2 \mathrm{m}$ without reducing the speed. The static friction coefficient is $0.2$. The cyclist will not slip and pass the curve $\left(g=9.8 \mathrm{m} {\mathrm{s}}^{-2}\right)$

Statement II : If the road is banked at an angle of $45°$, cyclist can cross the curve of $2 \mathrm{m}$ radius with the speed of $18.5 \mathrm{km} {\mathrm{h}}^{-1}$ without slipping. In the light of the above statements, choose the correct answer from the options given below.

A modern grand-prix racing car of mass $m$ is travelling on a flat track in a circular arc of radius $R$ with a speed $v.$ If the coefficient of static friction between the tyres and the track is ${\mu }_{\mathrm{s}},$ then the magnitude of negative lift $F_L$ acting downwards on the car is:

Two blocks ($m=0.5 \mathrm{kg}$ and $M=4.5 \mathrm{kg}$) are arranged on a horizontal frictionless table as shown in the figure. The coefficient of static friction between the two blocks is $\frac{3}{7}.$ Then the maximum horizontal force that can be applied on the larger block so that the blocks move together is $N.$ (Round off to the Nearest Integer) [Take $g$ as $9.8 \mathrm{m} {\mathrm{s}}^{-2}$]