Question

Water from a tap emerges vertically downwards with an initial velocity $V_{0}$ . Assuming pressure is constant throughout the stream of water and the flow is steady, find the distance from the tap at which cross-sectional area of the stream is half of the cross-sectional area of the stream at the tap.

• A. $\frac{V_{0}^{2}}{2 g}$
• B. $\frac{3 V_{0}^{2}}{2 g}$
• C. $\frac{2 V_{0}^{2}}{g}$
• D. $\frac{5 V_{0}^{2}}{2 g}$

Answer 1

Answer: (B)

Using the 3rd equation of motion
$V^{2}=u^{2}+2gh$
$V_{2}=\sqrt{V_{0}^{2} + 2 g h}$
Using the equation of continuity
$A_{1}V_{1}=A_{2}V_{2}$
we get $\frac{A_{2}}{A_{1}}=\frac{V_{0}}{\sqrt{V_{0}^{2} + 2 g h}}$
Required condition $\frac{A_{1}}{A_{2}}=\frac{1}{2}=\frac{V_{0}}{\sqrt{V_{0}^{2} + 2 g h}}$
$\Rightarrow 4V_{0}^{2}=V_{0}^{2}+2gh$
$h=\frac{3 V_{0}^{2}}{2 g}$