A cylinder containing water up to a height of 25 cm has a hole of cross-section $\frac{1}{4} {\mathrm{cm}}^2$ in its bottom. It is counterpoised in a balance. What is the initial change in the balancing weight when water begins to flow out

Let A = The area of cross section of the hole 
v = Initial velocity of efflux
d = Density of water, 
Initial volume of water flowing out per second = Av
Initial mass of water flowing out per second = Avd
Rate of change of momentum = Adv²     
Initial downward force on the flowing out water = Adv²
So equal amount of reaction acts upwards on the cylinder.  
$\therefore$ Initial upward reaction = ${\mathrm{Adv}}^2$   [As  $\mathrm{v}=\sqrt{2\mathrm{gh}}$ ]
$\therefore$ Initial decrease in weight  $=\mathrm{Ad}\left(2\mathrm{gh}\right)$
$=2\text{ Adgh }=2\times \left(\frac{1}{4}\right)\times 1\times 980\times 25=12.5 \mathrm{gm}-\mathrm{wt}.$