A bullet of mass $$200 \mathrm{~g}$$ having initial kinetic energy $$90 \mathrm{~J}$$ is shot inside a long swimming pool as shown in the figure. If it's kinetic energy reduces to $$40 \mathrm{~J}$$ within $$1 \mathrm{~s}$$, the minimum length of the pool, the bullet has to travel so that it completely comes to rest is

Assume there are two identical simple pendulum clocks. Clock - 1 is placed on the earth and Clock - 2 is placed on a space station located at a height h above the earth surface. Clock - 1 and Clock - 2 operate at time periods 4 s and 6 s respectively. Then the value of h is -

(consider radius of earth $$R_{E}=6400 \mathrm{~km}$$ and $$\mathrm{g}$$ on earth $$10 \mathrm{~m} / \mathrm{s}^{2}$$ )

Consider a cylindrical tank of radius $$1 \mathrm{~m}$$ is filled with water. The top surface of water is at $$15 \mathrm{~m}$$ from the bottom of the cylinder. There is a hole on the wall of cylinder at a height of $$5 \mathrm{~m}$$ from the bottom. A force of $$5 \times 10^{5} \mathrm{~N}$$ is applied an the top surface of water using a piston. The speed of ifflux from the hole will be : (given atmospheric pressure $$\mathrm{P}_{\mathrm{A}}=1.01 \times 10^{5} \mathrm{~Pa}$$, density of water $$\rho_{\mathrm{W}}=1000 \mathrm{~kg} / \mathrm{m}^{3}$$ and gravitational acceleration $$\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$$ )

A vessel contains $$14 \mathrm{~g}$$ of nitrogen gas at a temperature of $$27^{\circ} \mathrm{C}$$. The amount of heat to be transferred to the gas to double the r.m.s speed of its molecules will be :

Take $$\mathrm{R}=8.32 \mathrm{~J} \mathrm{~mol}^{-1} \,\mathrm{k}^{-1}$$.