Question

A garden hose with an internal diameter of $1.9cm$is connected to a (stationary) lawn sprinkler that consists merely of a container with $24$holes, each $0.13cm$in diameter. If the water in the hose has a speed of $0.91m/s$, at what speed does it leave the sprinkler holes?

Short answer

The speed of water through the sprinkler holes is $8.1m/s$.

Step-by-step solution

Given data

1. The diameter of a garden hose is

2. The number of holes of the lawn sprinkler is

3. The diameter of each hole of the sprinkler is

4. The speed of water in the hose is

Understanding the concept continuity equation

An ideal fluid is incompressible and lacks viscosity, and its flow is steady and irrotational. A streamline is a path followed by an individual fluid particle. A tube of flow is a bundle of streamlines. The flow within any tube of flow obeys the equation of continuity:

Which is the volume flow rate, is the cross-sectional area of the tube of flow at any point, and is the speed of the fluid at that point.

Write the continuity equation in terms of the diameter of each hole and the diameter of a garden hose. Then inserting the given values in it, find the speed of water through the sprinkler holes.

Formulae are as follows:

where A is area and v is velocity.

Determining the speed of water through the sprinkler holes

Let,A2be the area of a single hole of the sprinkler. Then, the equation of continuity becomes,

So,

Substituting this in the above equation,

Therefore, the speed of water through the sprinkler holes is