Water is flowing steadily in a river with a constant velocity gradient. A and B are the two layers of water at heights 40 cm and 90 cm from the bottom. The velocity of the layer A is 12 cm / s . The v

Question

Water is flowing steadily in a river with a constant velocity gradient. A and B are the two layers of water at heights 40 cm and 90 cm from the bottom. The velocity of the layer A is 12 cm / s . The velocity of the layer B is

TestHub · Accepted Answer

Since the velocity gradient is constant, $\frac{d v}{d y} = k$ (constant).

Integrating, we get $v = k y + c$ . At the bottom of the river ( $y = 0$ ), the velocity of water is zero ( $v = 0$ ), which gives $c = 0$ .

Thus, $v = k y$ , which implies $v \propto y$ .

$\frac{v _{A}}{y _{A}} = \frac{v _{B}}{y _{B}}$

Substituting the given values $y_{A} = 40 cm$ , $v_{A} = 12 cm/s$ , and $y_{B} = 90 cm$ :

$\frac{12}{40} = \frac{v _{B}}{90}$

$v_{B} = \frac{12 \times 90}{40} = 27 cm/s$

Answer: $27 cm / s$

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