Q129 / 160 · AP EAMCET 2020 (21 Sep Shift 2) · Physics

Water flows through a hose pipe whose internal diameter is \(4 \mathrm{~cm}\) at a speed of \(1 \mathrm{~ms}^{-1}\). If water has to emerge at a speed of \(4 \mathrm{~ms}^{-1}\), then the diameter of the nozzle should be

  • A
    \(1 \mathrm{~cm}\)
  • Correct answer
    B
    \(2 \mathrm{~cm}\)
  • C
    \(4 \mathrm{~cm}\)
  • D
    \(0.5 \mathrm{~cm}\)

Solution

Internal diameter for hose pipe, \(d_1=4 \mathrm{~cm}, r_1=\frac{d}{2}=\frac{4}{2}=2 \mathrm{~cm}=2 \times 10^{-2} \mathrm{~m}\) Speed of water through hose pipe, \(v_1=1 \mathrm{~ms}^{-1}\) Speed of water through nozzle, \(v_2=4 \mathrm{~ms}^{-1}\) Diameter, \(d_2=\) ? According to principle of continuity, \(\begin{array}{llll} & A_1 v_1=A_2 v_2 \\ \Rightarrow & \pi r_1^2 v_1=\pi r_2^2 v_2 \\ \Rightarrow & r_2^2=\frac{r_1^2 v_1}{v_2} \Rightarrow r_2^2=\frac{\left(2 \times 10^{-2}\right)^2 \times 1}{4} \\ \Rightarrow & r_2^2=10^{-4} \\ \Rightarrow & r_2=10^{-2} \mathrm{~m} \Rightarrow r_2=1 \mathrm{~cm} \end{array}\) \(\therefore\) Diameter of nozzle, \(d_2=2 r_2=2 \times 1=2 \mathrm{~cm}\)

Hence, the correct answer is option: 'B'.

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