$A$ car travelling at a speed of $30 \, km/h$ is brought to a halt in $8 \, m$ by applying brakes. If the same car is travelling at $60 \, km/h$,it can be brought to a halt with the same braking force in ............... $m$.

$A$ particle of mass $m$ is moving along the $x$-axis with initial velocity $u \hat{i}$. It collides elastically with a particle of mass $10m$ at rest and then moves with half its initial kinetic energy (see figure). If $\sin \theta_{1} = \sqrt{n} \sin \theta_{2}$,then the value of $n$ is:

$A$ particle $P$ is sliding down a frictionless hemispherical bowl. It passes the point $A$ at $t = 0$. At this instant of time, the horizontal component of its velocity is $v$. $A$ bead $Q$ of the same mass as $P$ is ejected from $A$ at $t = 0$ along the horizontal string $AB$ (see figure) with the speed $v$. Friction between the bead and the string may be neglected. Let ${t_P}$ and ${t_Q}$ be the respective time taken by $P$ and $Q$ to reach the point $B$. Then

$A$ train of mass $3 \times 10^6 \ kg$ is accelerated by an engine such that its velocity increases from $5 \ m/s$ to $25 \ m/s$ in $5 \ minutes$. The power of the engine is ........ $MW$.

$A$ pendulum of length $2\,m$ is released from point $P$ as shown in the figure. When it reaches point $Q$,it loses $10\%$ of its total energy due to air resistance. The velocity at $Q$ is .... $m/s$.

$A$ chain of length $L = 2 \ m$ is placed on a smooth table such that half of its length hangs over the edge. If the chain is released,find the velocity of the chain when it just leaves the table (in $m/s$). (Take $g = 10 \ m/s^2$)