In an elastic collision,

Two particles $A$ and $B$ of masses $2 \mathrm{kg}$ and $3 \mathrm{kg}$ moving on a smooth horizontal surface with speed $1.5 \mathrm{m} {\mathrm{s}}^{-1}$ and $2 \mathrm{m} {\mathrm{s}}^{-1}$collide perfectly inelastically, then

A particle strikes a smooth horizontal surface at an angle $\theta >0$ and it rebounds at an angle $\varphi$ as shown

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

Consider a vertical tube open at both ends. The tube consists of two parts, each of different cross-sections and each part having a piston which can move smoothly in respective tubes. The two pistons are joined together by an inextensible wire. The combined mass of the two pistons is $5 \mathrm{kg}$and area of cross-section of the upper piston is $10 {\mathrm{cm}}^2$greater than that of the lower piston. The amount of gas enclosed by the pistons is one mole. When the gas is heated slowly, the pistons moves by $50 \mathrm{cm}$ as shown in figure. The rise in temperature of the gas is in the form $\frac{X}{R} \mathrm{K}$, where $R$ is universal gas constant. Use, $g=10 \mathrm{m} {\mathrm{s}}^{-2}$ and outside pressure$={10}^5 \mathrm{N} {\mathrm{m}}^{-2}$. Fill value of $X$.