A particular garden hose has an inside diameter of 15/16 in. The nozzle opening has a diameter of 5/16 in. When the flow speed of the water in the hose is 0.58 m/s, what is the flow speed through the nozzle?
To find the flow speed through the nozzle, we can use the principle of conservation of mass. According to this principle, the flow rate of water at any point in a pipe or hose remains constant, assuming the pipe or hose is of uniform cross-section.
The flow rate is given by the equation:
Flow rate = (cross-sectional area) × (flow speed)
Let's denote the flow speed through the nozzle as V_n and the flow speed in the hose as V_h. We need to find V_n.
We know that the diameter of the hose is 15/16 inches, which means the radius is (15/16) / 2 inches. To convert this to meters, we can use the conversion factor 1 inch = 0.0254 meters.
So, the radius of the hose (r_h) becomes:
r_h = ((15/16) / 2) * 0.0254 meters
We can apply the same process to find the radius of the nozzle (r_n) using the diameter of 5/16 inches.
Now, the flow rate in the hose (Q_h) is equal to the flow rate in the nozzle (Q_n):
Q_h = Q_n
By substituting the formulas for flow rate, we get:
(cross-sectional area of hose) × V_h = (cross-sectional area of nozzle) × V_n
The cross-sectional area of a circle is given by the formula:
Area = π × (radius)^2
Substituting the values, we have:
(π × r_h^2) × V_h = (π × r_n^2) × V_n
Using the given flow speed in the hose (V_h = 0.58 m/s), we can now solve for V_n.
1. Calculate the radius of the hose (r_h) using the diameter of 15/16 inches.
2. Calculate the radius of the nozzle (r_n) using the diameter of 5/16 inches.
3. Substitute the values of r_h, r_n, and V_h into the equation: (π × r_h^2) × V_h = (π × r_n^2) × V_n
4. Solve for V_n by rearranging the equation: V_n = ((π × r_h^2) × V_h) / (π × r_n^2)
5. Simplify the equation to get the final value of V_n.