Question

Water exits a garden hose at a speed of 1.2 m/s. If the end of the garden hose is 1.5 cm in diameter and you want to make the water go 15 m high, what fraction of the area of the hole do you have to block off with your thumb?A : 93%B : 14%C : 7.0%D : 73%E : 27%

Answer 1

A 93%

Step-by-step explanation:

`P_1=P_2` = Pressure will be equal at inlet and outlet

`\rho` = Density of water = 1000 kg/m³

g = Acceleration due to gravity = 9.81 m/s²

`v_1` = Velocity at inlet = 1.2 m/s

`v_2` = Velocity at outlet

`r_1` = Radius of inlet =
`(1.5)/(2)=0.75\ cm`

`r_2` = Radius of outlet

From Bernoulli's relation

`P_1+(1)/(2)\rho v_1^2+\rho gh_1=P_2+(1)/(2)\rho v_2^2+\rho gh_2\\\Rightarrow (1)/(2)\rho v_1^2+\rho gh_1=(1)/(2)\rho v_2^2+\rho gh_2\\\Rightarrow v_2=\sqrt{2((1)/(2)v_1^2+gh_1-gh_2)}\\\Rightarrow v_2=\sqrt{2((1)/(2)1.2^2+9.81* 15)}\\\Rightarrow v_2=17.19709\ m/s`

From continuity equation

`A_1v_1=A_2v_2\\\Rightarrow \pi r_1^2v_1=\pi r_2^2v_2\\\Rightarrow r_2=\sqrt{(r_1^2v_1)/(v_2)}\\\Rightarrow r_2=\sqrt{(0.0075^2* 1.2)/(17.19709)}\\\Rightarrow r_2=0.00198\ m`

The fraction would be

`(A_1-A_2)/(A_1)* 100=(r_1^2-r_2^2)/(r_1^2)* 100\\ =(0.0075^2-0.00198^2)/(0.0075^2)* 100\\ =93.0304\ \%`

The fraction is 93.0304%