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Streamline Flow

Water from a ...

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 stream is half of the cross-sectional area of stream at the top:

V_{0}^{2} / 2 g

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3 V_{0}^{2} / 2 g

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2 V_{0}^{2} / g

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5 V_{0}^{2} / 2 g

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Solution

The correct option is A $3 V_{0}^{2} / 2 g$
velocity equation $V_{2}^{2} = V_{0}^{2} + 2 g h$ and continuity equation $A_{1} V_{0} = A_{2} V_{2}$ .....(i)
Solving, we get $\frac{A_{2}}{A_{1}} = \frac{V_{0}}{\sqrt{V_{0}^{2} + 2 g h}}$
$A_{2} / A_{1} = \frac{1}{2} = \frac{V_{0}}{\sqrt{V_{0}^{2} + 2 g h}}$
$4 V_{0}^{2} = V_{0}^{2} + 2 g h$
$h = \frac{3 V_{0}^{2}}{2 g}$

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