A water clock (clepsydra) used in a ancient Greece is designed as a vessel with a small orifice O( figure).The time is determined according to the level of the water in the vessel as show in figure. The equation corresponding to the shape of the vessel for the time scale to uniform is  $y=k{x}^{4}wherek=\frac{{\pi }^2{v}^2}{pg{a}^2}$. Value of p will be ? Assume vessel is axially symmetrical and O is origin. 
$v$ : Speed of  upper layer; g : Acceleration due gravity; a: area of cross section of Orifice  

Let us introduce the coordinate system depicted in figure. According to Torricelli’s formula, the outflow velocity of a liquid is  $V=\sqrt{2gy}$,where y is the thickness of the water layer in the upper vessel, Since water is incompressible  $aV=Av$ ,where  $v$  is the velocity with which the upper layer on the water lowers, A is its area, and  $a$  is the area of the orifice.
 
If we assume that the vessel is axially symmetrical, then  $A=\pi x^2$, where  $x$  is the horizontal coordinate of the vessel wall. Therefore  $\frac{\pi x^2}{\sqrt{2gy}}=\frac{a}{v}=\text{cost}$ 
Since in conformity with the initial condition, the water level should lower with a constant velocity. Hence, the shape of the vessel can be determined from the equation  $y=k{x}^4$
Where  $k=\frac{{\pi }^2{v}^2}{2g{a}^2}$