$A$ vessel containing water is moving with a constant speed towards the right along a straight horizontal path. Which of the following diagrams represents the surface of the liquid?

$(a)$ It is known that the density $\rho$ of air decreases with height $y$ as $\rho = \rho_{0} e^{-y / y_{0}}$,where $\rho_{0} = 1.25 \; kg \, m^{-3}$ is the density at sea level,and $y_{0}$ is a constant. This density variation is called the law of atmospheres. Obtain this law assuming that the temperature of the atmosphere remains constant (isothermal conditions). Also,assume that the value of $g$ remains constant.
$(b)$ $A$ large $He$ balloon of volume $1425 \; m^{3}$ is used to lift a payload of $400 \; kg$. Assume that the balloon maintains a constant radius as it rises. How high does it rise?
[Take $y_{0} = 8000 \; m$ and $\rho_{He} = 0.18 \; kg \, m^{-3}$]

An air bubble of volume $50\, cm^3$ is formed at the bottom of a lake $47.6\, m$ deep. What will be its volume when it reaches the surface? (Atmospheric pressure $= 70\, cm$ of $Hg$ and density of $Hg = 13.6\, g/cm^3$)

$A$ uniform meter scale is supported at its $20 \ cm$ mark. $A$ body suspended from the $10 \ cm$ mark keeps the scale horizontal. However,the scale becomes unbalanced if the body is completely immersed in water. To regain the balance,the body is shifted to the $8 \ cm$ mark. Therefore,the specific gravity of the material of the body is

$A$ razor-blade floats on the surface of water contained in a glass. When the glass is gently shaken,the razor-blade sinks. Mark the incorrect statement.

$A$ river of salty water is flowing with a velocity $2 \,m/s$. If the density of the water is $1.2 \,g/cc$, then the kinetic energy of each cubic metre of water is