An air bubble of volume $1\,cm^3$ rises from the bottom of a lake $40\,m$ deep to the surface at a temperature of $12^{\circ}C$. The atmospheric pressure is $1 \times 10^5\,Pa$,the density of water is $1000\,kg/m^3$,and $g = 10\,m/s^2$. There is no difference in the temperature of water at the depth of $40\,m$ and on the surface. The volume of the air bubble when it reaches the surface will be $..........\,cm^3$. (in $,cm^3$)

$A$ pump is used to deliver water at a certain rate through a pipe. To obtain $n$ times the water in the same time,by what factor should the velocity of water,the force of water,and the power of the pump be increased?

At a speed of .......... $m/s$,the velocity head of water is equal to the pressure head of $40\, cm$ of $Hg$.

An air bubble rises from the bottom of a lake to the surface. If its radius increases by $200 \%$ and the atmospheric pressure is equal to a water column of height $H$,then the depth of the lake is ..... $H$.

$A$ spray gun is shown in the figure where a piston pushes air out of a nozzle. $A$ thin tube of uniform cross section is connected to the nozzle. The other end of the tube is in a small liquid container. As the piston pushes air through the nozzle,the liquid from the container rises into the nozzle and is sprayed out. For the spray gun shown,the radii of the piston and the nozzle are $20 \ mm$ and $1 \ mm$ respectively. The upper end of the container is open to the atmosphere.
$1.$ If the piston is pushed at a speed of $5 \ mm \ s^{-1}$,the air comes out of the nozzle with a speed of
$(A)$ $0.1 \ m \ s^{-1}$ $(B)$ $1 \ m \ s^{-1}$ $(C)$ $2 \ m \ s^{-1}$ $(D)$ $8 \ m \ s^{-1}$
$2.$ If the density of air is $\rho_{a}$ and that of the liquid is $\rho_{\ell}$,then for a given piston speed,the rate (volume per unit time) at which the liquid is sprayed will be proportional to
$(A)$ $\sqrt{\frac{\rho_{a}}{\rho_{\ell}}}$ $(B)$ $\sqrt{\rho_{a} \rho_{\ell}}$ $(C)$ $\sqrt{\frac{\rho_{\ell}}{\rho_{a}}}$ $(D)$ $\rho_{\ell}$

The Karman line is a theoretical construct that separates the Earth's atmosphere from outer space. It is defined as the height at which the lift on an aircraft flying at the speed of a polar satellite $(8 \, km/s)$ is equal to its weight. Taking a fighter aircraft of wing area $30 \, m^2$ and mass $7500 \, kg$,the height of the Karman line above the ground will be in the range of .............. $km$. (Assume the density of air at height $h$ above the ground to be $\rho(h) = 1.2 e^{-h/10} \, kg/m^3$,where $h$ is in $km$,and the lift force to be $\frac{1}{2} \rho v^2 A$,where $v$ is the speed of the aircraft and $A$ is its wing area.)