Question

A tank of large base area is filled with water up to a height of $5m$. A hole of $2c{m}^2$ cross section section in the bottom allows the water to drain out in continuous streams. For this situation, mark out the correct statement(s) (take ${\rho }_{water}=1000kg{/}^2,g=10m/s^2)$.

• A. The cross sectional area of the emerging stream of water decreases as it falls down
• B. The cross sectional area of the emerging stream of water increases as it falls down
• C. At a distance of $5m$ below the bottom of the tank, the cross-sectional area of the stream is $1.414c{m}^2$
• D. At a distance of $5m$ below the bottom of the tank, the cross-sectional area of the stream is $2.86c{m}^2$

Answer 1

Answer: (A), (C)

The correct options are
A The cross sectional area of the emerging stream of water decreases as it falls down
C At a distance of $5m$ below the bottom of the tank, the cross-sectional area of the stream is $1.414c{m}^2$

As the water falls down in the stream, the speed of the stream will increase due to gravity. So according to equation of continuity $(av=\text{a constant})$, the area of the stream will decrease to keep the product of area and speed constant. $\Rightarrow$ option A is correct.
Velocity of water just coming out of the hole is $v_1=\sqrt{2gh}=\sqrt{2\times 9.8\times 5}=7\sqrt{2}\text{m/s}$
Velocity of water as it falls down 5m more
$v_2=\sqrt{(7\sqrt{2}{)}^2+2\times 9.8\times 5}=14\text{m/s}$ $\because v_2=\sqrt{{v_1}^2+2gh}$
Now using the equation of continuity, we have
$a_1{v}_1=a_2{v}_2\Rightarrow 2\times 7\sqrt{2}=a_2\times 14$
$\Rightarrow a_2=\sqrt{2}{\text{cm}}^2=1.414{\text{cm}}^2$ $\Rightarrow$ option C is correct.