Water flowing through a garden hose of diameter 2.73 cm fills a 25.0-L bucket in 1.20 min.
(a) What is the speed of the water leaving the end of the hose?
m/s
(b) A nozzle is now attached to the end of the hose. If the nozzle diameter is one-third the diameter of the hose, what is the speed of the water leaving the nozzle?
m/s
To find the speed of the water leaving the end of the hose, we can use the equation for the volumetric flow rate of water.
The volumetric flow rate (Q) is given by the formula:
Q = A * v
Where:
Q is the volumetric flow rate (in m³/s),
A is the cross-sectional area of the hose or nozzle (in m²),
v is the velocity of the water (in m/s).
(a) To find the speed of the water leaving the end of the hose, we'll calculate the cross-sectional area of the hose using its diameter.
The diameter of the hose is given as 2.73 cm, which is equal to 0.0273 m. The radius (r) of the hose is half the diameter, so r = 0.01365 m.
To calculate the cross-sectional area (A) of the hose, we use the formula:
A = π * r^2
Substituting the values, we get:
A = π * (0.01365)^2
Now, we can find the volumetric flow rate (Q) using the following formula:
Q = (Volume / Time)
Given that the volume of water that fills the 25.0-L bucket in 1.20 min, we need to first convert the volume to cubic meters (m³) and the time to seconds (s).
Volume = 25.0 L = 0.0250 m³ (since 1 L = 0.001 m³)
Time = 1.20 min = 72 s (since 1 min = 60 s)
Therefore, the volumetric flow rate (Q) is:
Q = 0.0250 m³ / 72 s
Now let's substitute the calculated values for A and Q into the equation Q = A * v:
0.0250 m³ / 72 s = π * (0.01365)^2 * v
Solving for v, we get the speed of the water leaving the end of the hose.
(b) Now let's calculate the speed of the water leaving the nozzle, where the diameter of the nozzle is one-third of the hose diameter.
The diameter of the nozzle = (1/3) * 2.73 cm = 0.91 cm = 0.0091 m
Using the same method as in part (a), we find the cross-sectional area (A) of the nozzle:
A = π * (0.00455)^2
And we already have the volumetric flow rate (Q) from part (a).
0.0250 m³ / 72 s = π * (0.00455)^2 * v
Solving for v, we get the speed of the water leaving the nozzle.
By following these steps, you can calculate the speed of water leaving both the hose and the nozzle.