Question

A water hose 2cm in diameter is used to fill a 20 litre bucket. If it takes 1 minute to fill the bucket, then the speed v at which the water leaves the hose is?

Answer 1

Volumetric rate of flow (Q) can be calculated as a product of Area (A) and velocity (v). Here, area is calculated using the given diameter. Also, the volumetric rate is calculated by dividing volume by time.
Formula used:
Volumetric rate of flow is given by,
$Q=vA$
where, v is velocity and A is area of cross section.

Complete answer:
Given, diameter = 2 cm.
Therefore, area is calculated as:
$\text{A=}\dfrac{\pi {{d}^{2}}}{4}=\dfrac{\pi }{4}{{\left( 0.02 \right)}^{2}}$
Volume rate of flow of liquid (Q) is calculated as:
$Q=\dfrac{volume}{time}=\dfrac{20L}{60}$
$\Rightarrow Q=\dfrac{{{10}^{-3}}}{3}{{m}^{3}}{{s}^{-1}}$
Now,
$Q=vA$
Therefore, velocity is calculated as:
$v=\dfrac{Q}{A}$
$\Rightarrow v=\dfrac{{{10}^{-3}}}{3}\times \dfrac{4}{\pi {{\left( 0.002 \right)}^{2}}}$
$\Rightarrow v=1.06m{{s}^{-1}}$

Note:
Equation of continuity is better used in Bernoulli's Theorem. We use this theorem as the same as the law of conservation of energy of the fluid. For a small area of cross-section velocity increases and for a large area velocity of fluid decreases to maintain mass conservation according to
${{A}_{1}}{{v}_{1}}={{A}_{2}}{{v}_{2}}=\text{constant}$