Question

Two children, Ferdinand and Isabella, are playing with a water hose on a sunny summer day. Isabella is holding the hose in her hand 1.0 meters above the ground and is trying to sprayFerdinand, who is standing 10.0 meters away.

Part A

Will Isabella be able to spray Ferdinand if the water is flowing out of the hose at a constant speed v₀ of 3.5 meters per second? Assume that the hose is pointed parallel to the ground and take the magnitude of the acceleration g due to gravity to be 9.81 meters per second, per second.

• Yes
• No

Part B

To increase the range of the water, Isabella places her thumb on the hose hole and partially covers it. Assuming that the flow remains steady, what fraction f of the cross-sectional area of the hose hole does she have to cover to be able to spray her friend?

Assume that the cross-section of the hose opening is circular with a radius of 1.5 centimeters.

Express your answer as a percentage to the nearest integer.

 

 

Answer 1

According to the equations of motion,

\(y=y_{0}+u t-\frac{1}{2} g t^{2}\)

\(y_{0}=\frac{1}{2} g t^{2}\)

\(t=\sqrt{\frac{2 y_{0}}{g}}\)

\(=\sqrt{\frac{2 \times 1}{9.81}}\)

\(t=0.4515 \mathrm{sec}\)

The horizontal distance covered by the water is

\(x=x_{0}+v t\)

\(10=v t\)

\(v=\frac{10}{t}\)

\(=\frac{10}{0.4515}\)

\(v=22.1 \mathrm{~m} / \mathrm{s}\)

The water flowing out of the hose at a constant speed of \(\mathrm{v}_{0}=3.5 \mathrm{~m} / \mathrm{s}\) The cross section area of the larger section is \(A_{0}=\pi r^{2}\)

\(=\pi \times\left(1.5 \times 10^{-2}\right)^{2}\)

\(=7.068 \times 10^{-4} \mathrm{~m}^{2}\)

According to the continuity equation,

\(A v=A_{0} v_{0}\)

\(A=\frac{A_{0} v_{0}}{v}\)

\(=\frac{7.068 \times 10^{-4} \times 3.5}{22.1}\)

\(A=1.12 \mathrm{~cm}^{2}\)

Thus the fraction is

\(f=\frac{A_{0}-A}{A_{0}}\)

\(=\frac{7.068 \times 10^{-4}-1.12 \times 10^{-4}}{7.068 \times 10^{-4}}\)

\(=0.84\)

\(f=84 \%\)

A) NO

B) f = 84%