Answer:
(a).The speed of the water in the nozzle is 3.014 m/s.
(b). The pressure in the nozzle is 1.86 atm.
Explanation:
Given that,
Nozzle diameter = 0.25 in = 0.00635 m
Hose pipe diameter = 0.64 in = 0.016256 m
Pressure = 1.9 atm =192518 Pa
(a). We need to calculate the speed of the water in the nozzle
Flow Speed at the inlet pipe will be given by using Continuity Equation
![Q_{1}=Q_{2}](https://tex.z-dn.net/?f=Q_%7B1%7D%3DQ_%7B2%7D)
![v_{1}A_{1}=v_{2}A_{2}](https://tex.z-dn.net/?f=v_%7B1%7DA_%7B1%7D%3Dv_%7B2%7DA_%7B2%7D)
![v_{1}=v_{2}\times(\dfrac{A_{2}}{A_{1}})](https://tex.z-dn.net/?f=v_%7B1%7D%3Dv_%7B2%7D%5Ctimes%28%5Cdfrac%7BA_%7B2%7D%7D%7BA_%7B1%7D%7D%29)
Where, A = area of pipe
![A=\pi\times \dfrac{d^2}{4}](https://tex.z-dn.net/?f=A%3D%5Cpi%5Ctimes%20%5Cdfrac%7Bd%5E2%7D%7B4%7D)
![v_{1}=v_{2}\times(\dfrac{d_{2}^2}{d_{1}^2})](https://tex.z-dn.net/?f=v_%7B1%7D%3Dv_%7B2%7D%5Ctimes%28%5Cdfrac%7Bd_%7B2%7D%5E2%7D%7Bd_%7B1%7D%5E2%7D%29)
Put the value into the formula
![v_{1}=0.46\times\dfrac{(0.016256)^2}{(0.00635)^2}](https://tex.z-dn.net/?f=v_%7B1%7D%3D0.46%5Ctimes%5Cdfrac%7B%280.016256%29%5E2%7D%7B%280.00635%29%5E2%7D)
![v_{1}=3.014\ m/s](https://tex.z-dn.net/?f=v_%7B1%7D%3D3.014%5C%20m%2Fs)
The speed of the water in the nozzle is 3.014 m/s.
(b). We need to calculate the pressure in the nozzle
Using Bernoulli's Theorem,
![P_{1}+\dfrac{1}{2}\rho\times v_{1}^2+\rho gh_{1}=P_{2}+\dfrac{1}{2}\rho\times v_{2}^2+\rho gh_{2}](https://tex.z-dn.net/?f=P_%7B1%7D%2B%5Cdfrac%7B1%7D%7B2%7D%5Crho%5Ctimes%20v_%7B1%7D%5E2%2B%5Crho%20gh_%7B1%7D%3DP_%7B2%7D%2B%5Cdfrac%7B1%7D%7B2%7D%5Crho%5Ctimes%20v_%7B2%7D%5E2%2B%5Crho%20gh_%7B2%7D)
Where, ![h_{1}=h_{2}](https://tex.z-dn.net/?f=h_%7B1%7D%3Dh_%7B2%7D)
![P_{1}+\dfrac{1}{2}\rho\times v_{1}^2=P_{2}+\dfrac{1}{2}\rho\times v_{2}^2](https://tex.z-dn.net/?f=P_%7B1%7D%2B%5Cdfrac%7B1%7D%7B2%7D%5Crho%5Ctimes%20v_%7B1%7D%5E2%3DP_%7B2%7D%2B%5Cdfrac%7B1%7D%7B2%7D%5Crho%5Ctimes%20v_%7B2%7D%5E2)
![P_{1}=P_{2}+\dfrac{1}{2}\rho(v_{2}^2-v_{1}^2)](https://tex.z-dn.net/?f=P_%7B1%7D%3DP_%7B2%7D%2B%5Cdfrac%7B1%7D%7B2%7D%5Crho%28v_%7B2%7D%5E2-v_%7B1%7D%5E2%29)
Put the value into the formula
![P_{1}=192518 +\dfrac{1}{2}\times1000\times((0.46)^2-(3.014)^2)](https://tex.z-dn.net/?f=P_%7B1%7D%3D192518%20%2B%5Cdfrac%7B1%7D%7B2%7D%5Ctimes1000%5Ctimes%28%280.46%29%5E2-%283.014%29%5E2%29)
![P_{1}=188081.702\ Pa](https://tex.z-dn.net/?f=P_%7B1%7D%3D188081.702%5C%20Pa)
![P=1.86\ atm](https://tex.z-dn.net/?f=P%3D1.86%5C%20atm)
Hence, (a).The speed of the water in the nozzle is 3.014 m/s.
(b). The pressure in the nozzle is 1.86 atm.