Question

Question: You are watering your lawn with a hose when you put your finger over the hose opening to increase the distance the water reaches. If you are holding the hose horizontally, and the distance the water reaches increases by a factor of 4, what fraction of the hose opening did you block?

Short answer

The fraction by which the hose opening is blocked is \(\left( {\frac{1}{2}} \right)\).

Step-by-step solution

Given Data

The increase in range of the water is by a factor of 4.

Understanding the continuity equation

In this problem, the continuity equation will be applied for evaluating the fraction of the hose opening. Initially, estimate the relation between the range of the water and speed of the water.

Calculating the fraction of the hose opening blocked

The level range formulais given by,

\(R = \frac{{{v^2}\sin 2\theta }}{g}\)

Here,\(R\)is the range,\(v\)is the speed,\(\theta \)is the angle and\(g\)is the gravitational acceleration.

From the above relation, the increase in the value of range by a factor of 4, rises the value of speed by 2 times.

The relation from continuity equation is given by,

\(\begin{array}{c}{A_1}{v_0} = {A_2}v\\{A_2} = \frac{{{A_1}{v_0}}}{v}\end{array}\)

Here,\({A_1}\)and\({A_2}\)are the area at inlet and outlet.

When the speed increases by 2 times, the area of the hose will become,

\(\begin{array}{l}{A_2} = \frac{{{A_1}{v_0}}}{{2{v_0}}}\\{A_2} = \left( {\frac{1}{2}} \right){A_1}\end{array}\)

Thus, the fraction by which the hose opening is blocked is \(\left( {\frac{1}{2}} \right)\).