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

Hint: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
\[{{A}_{1}}{{v}_{1}}={{A}_{2}}{{v}_{2}}=\text{constant}\]