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Chapter 15: Problem 36

A fire hose \(10 \mathrm{cm}\) in diameter delivers water at \(15 \mathrm{kg} / \mathrm{s}\). The hose terminates in a 2.5 -cm-diameter nozzle. What are the flow speeds (a) in the hose and (b) at the nozzle?

Short Answer

Expert verified
(a) The flow speed in the hose is approximately \( 0.6 m/s \). (b) The flow speed at the nozzle is approximately \( 9.6 m/s \).

Step by step solution

01

Identify Known Values

We are given that the hose diameter is \( 10 cm \), the nozzle diameter is \( 2.5 cm \), and the mass flow rate is \( 15 kg/s \). The mass density of water \( \rho \) is \( 1 g/cm^3 = 1000 kg/m^3 \). We convert the diameters to meters to ensure the units match when calculating the area.
02

Calculate Area of Hose and Nozzle

We calculate the area of the hose and the nozzle using the formula of the area of a circle \( A=\pi r^{2} \). The radius is half the diameter. The area of the hose \( A_{1} \) is \( \pi (\frac{10}{200})^{2} m^2 \), and the area of the nozzle \( A_{2} \) is \( \pi (\frac{2.5}{200})^{2} m^2 \).
03

Calculate the Flow Speed in the Hose

We can calculate the flow speed in the hose by using the equation for mass flow rate \( \rho A_{1} v_{1} = constant \). Solving for \( v_1 \), we get \( v_{1} = \frac{constant}{\rho A_{1}} \). We plug in the given constant mass flow rate (\( 15 kg/s \)), the mass density of water, and the calculated hose area to get the flow speed in the hose.
04

Calculate the Flow Speed at the Nozzle

Similarly, we calculate the flow speed at the nozzle by solving the mass flow rate equation for \( v_{2} \). We get \( v_{2} = \frac{constant}{\rho A_{2}} \). We plug in the given mass flow rate, the mass density of water, and the calculated nozzle area to get the flow speed in the nozzle.

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