State and establish principle of conservation of mechanical energy.

A particle $(\mathrm{m}=1\; \mathrm{kg})$ slides down a frictionless track $(AOC)$ starting from rest at a point $A$ (height $2\; \mathrm{m}$ ). After reaching $\mathrm{C}$, the particle continues to move freely in air as a projectile. When it reaching its highest point $P$ (height $1 \;\mathrm{m}$ ). the kinetic energy of the particle (in $\mathrm{J}$ ) is : (Figure drawn is schematic and not to scale; take $\left.g=10 \;\mathrm{ms}^{-2}\right)$

$A$ block of mass $m$ starts from rest and slides down $a$ frictionless semi-circular track from $a$ height $h$ as shown. When it reaches the lowest point of the track, it collides with a stationary piece of putty also having mass $m$. If the block and the putty stick together and continue to slide, the maximum height that the block-putty system could reach is:

An object flying in alr with velocity $(20 \hat{\mathrm{i}}+25 \hat{\mathrm{j}}-12 \hat{\mathrm{k}})$ suddenly breaks in two pleces whose masses are in the ratio $1: 5 .$ The smaller mass flies off with a velocity $(100 \hat{\mathrm{i}}+35 \hat{\mathrm{j}}+8 \hat{\mathrm{k}}) .$ The velocity of larger piece will be

$Assertion$ : A helicopter must necessarily have two propellers.
$Reason$ : Two propellers are provided in helicopter in order to conserve linear momentum

Three balls, $A, B$ and $C$ are released and all reach the point $X$ (shown in the figure). Balls $A$ and $B$ are released from two identical structures, one kept on the ground and the other at height $h$, from the ground as shown in the figure. They take time $t_A$ and $t_B$ respectively to reach $X$ (time starts after they leave the end of the horizontal portion of the structure). The ball $C$ is released from a point at height $h$, vertically above $X$ and reaches $X$ in time $t_C$. Choose the correct option.