Water flows with a speed of 0.530 m/s through a hose with a diameter of 3.30 cm . If the hose is attached to a nozzle with a diameter of 0.802 cm , what is the speed of water in the nozzle?
To find the speed of water in the nozzle, we can use the principle of conservation of mass. According to this principle, the flow rate of water remains constant throughout the hose, regardless of the change in diameter.
The flow rate of water can be calculated using the equation:
Q = A * v
where:
Q is the flow rate
A is the cross-sectional area of the hose or nozzle
v is the speed of water
The cross-sectional area can be calculated using the equation:
A = π * r^2
where:
π is a mathematical constant approximately equal to 3.14159
r is the radius of the hose or nozzle
Let's calculate the flow rate at the hose first.
Radius of the hose:
r_hose = diameter_hose / 2 = 3.30 cm / 2 = 1.65 cm = 0.0165 m
Cross-sectional area of the hose:
A_hose = π * r_hose^2 = 3.14159 * (0.0165 m)^2 = 0.00085 m^2
Flow rate at the hose:
Q_hose = A_hose * v_hose
Since the flow rate is constant, Q_hose = Q_nozzle.
Now let's calculate the cross-sectional area and speed at the nozzle.
Radius of the nozzle:
r_nozzle = diameter_nozzle / 2 = 0.802 cm / 2 = 0.401 cm = 0.00401 m
Cross-sectional area of the nozzle:
A_nozzle = π * r_nozzle^2 = 3.14159 * (0.00401 m)^2 = 0.00005 m^2
Flow rate at the nozzle:
Q_nozzle = A_nozzle * v_nozzle
Since the flow rate is constant, we can set it equal to the flow rate at the hose:
Q_hose = Q_nozzle
A_hose * v_hose = A_nozzle * v_nozzle
Solving for v_nozzle:
v_nozzle = (A_hose * v_hose) / A_nozzle
Substituting the given values:
v_nozzle = (0.00085 m^2 * 0.530 m/s) / 0.00005 m^2
Calculating the result:
v_nozzle = 9.13 m/s
Therefore, the speed of water in the nozzle is approximately 9.13 m/s.
To find the speed of water in the nozzle, we can use the principle of conservation of mass, which states that the mass of a fluid entering a section of a pipe is equal to the mass leaving that section. Since density is constant, we can also say that the volume entering the section is equal to the volume leaving the section.
The volume flow rate is given by the formula:
Q = Av
Where Q is the volume flow rate, A is the cross-sectional area of the pipe, and v is the velocity of the fluid.
Given that the hose has a diameter of 3.30 cm, we can calculate its cross-sectional area using the formula:
A = π * (d/2)^2
Where A is the cross-sectional area and d is the diameter of the hose.
Let's calculate the cross-sectional area of the hose:
A_hose = π * (3.30 cm/2)^2
Now, let's calculate the cross-sectional area of the nozzle using the same formula:
A_nozzle = π * (0.802 cm/2)^2
Since the volume entering the hose is equal to the volume leaving the nozzle, we can set up the equation:
Q_hose = Q_nozzle
A_hose * v_hose = A_nozzle * v_nozzle
Since we are solving for v_nozzle, we rearrange the equation:
v_nozzle = (A_hose / A_nozzle) * v_hose
Now, we substitute the known values:
v_nozzle = (A_hose / A_nozzle) * 0.530 m/s
Finally, we calculate the speed of water in the nozzle:
v_nozzle = (A_hose / A_nozzle) * 0.530 m/s
v_nozzle = [(π * (3.30 cm/2)^2) / (π * (0.802 cm/2)^2)] * 0.530 m/s
Using the calculated values, we can find the speed of water in the nozzle.