Question

Water flows through a hose (pipe), whose internal diameter is \[2.0\,m\] at a speed of \[1.0\,m.{s^{ - 1}}\]. What should be the diameter of the nizzle, if the water is to emerge at a speed of \[4.0\,m.{s^{ - 1}}\]
A. \[1\,m\]
B. \[1\,cm\]
C. \[0.005\,m\]
D. \[0.5\,m\]

Answer 1

Hint: We have to know about the continuity equation to solve this question. For a control volume that has a solitary gulf and a solitary outlet, the rule of preservation of mass expresses that, for a consistent state stream, the mass stream rate into the volume should approach the mass stream rate out.

Formula used:
\[{a_1}{v_1} = {a_2}{v_2}\]
Here ${a_1}$, ${a_2}$ are the area of cross section and ${v_1}$, ${v_2}$ are the rate of flow of volume of a liquid.

Complete step by step answer:
The continuity equation describes the transport of some quantities like fluid or gas. The equation explains how a fluid conserves mass in its motion. Many physical phenomena like energy, mass, momentum, natural quantities, and electric charge are conserved using the continuity equations. We know that the continuity equation is basically a numerical articulation of the rule of protection of mass. From continuity equation we can write,
\[{a_1}{v_1} = {a_2}{v_2}\]
\[ \Rightarrow \pi {(1)^2} \times 1 = 4 \times \pi {r^2}\]
\[ \Rightarrow \dfrac{1}{4} = {r^2} \\
\therefore r = \dfrac{1}{2}cm\]
Here $r$ is the radius of the nozzle and we know that the diameter is half of radius. So, diameter of the nozzle is equal to \[\dfrac{1}{2} \times 2\,cm = 1\,cm\]

Hence, the correct option is B.

Note: We can also say that, a diameter of a circle is any straight line fragment that goes through the focal point of the circle and whose endpoints lie on the circle. It can likewise be characterized as the longest harmony of the circle. The two definitions are likewise legitimate for the breadth of a circle. In more current use, the length of a distance across is additionally called the breadth. In this sense one discusses the distance across instead of a measurement (which alludes to the line section itself), since all breadths of a circle or circle have a similar length, this being double the sweep r. we have to keep these in our mind.