Final answer:
The question asks to calculate the flow rate of water through a hose and the diameter of the nozzle based on given velocities. The flow rate is calculated using the area of the hose and the velocity of the water. Applying the conservation of mass, the diameter of the nozzle is found by equating the flow rate in the hose to the flow rate in the nozzle.
Step-by-step explanation:
The question involves applying principles of fluid dynamics to calculate the water flow rate through a hose and the diameter of the nozzle based on the change in water velocity. To find the flow rate in liters per second, we use the formula: flow rate (Q) = cross-sectional area (A) × velocity (v).
The cross-sectional area of a hose with a diameter (D) is given by A = π(D/2)^2. So for a hose with an internal diameter of 1.85 cm and water moving at a speed of 2.25 m/s, the flow rate Q = π(0.0185 m/2)^2 × 2.25 m/s. Converting cubic meters to liters (1 m^3 = 1000 L), we find the flow rate in liters per second. For part (b) of the question, we apply the principle of conservation of mass, which dictates that the flow rate must remain constant.
Hence, the velocity increase in the nozzle means that the nozzle's cross-sectional area must decrease. By setting the flow rate at the hose equal to the flow rate at the nozzle, we can solve for the nozzle's diameter. Given the faster velocity of 15.0 m/s, we can find the necessary smaller diameter in centimeters.
To apply the formula, we set Q_hose = Q_nozzle:
A_hose × v_hose = A_nozzle × v_nozzle
Since the area is proportional to the square of the diameter, this can also be written as:
(D_hose^2 × v_hose = D_nozzle^2 × v_nozzle)
By knowing the hose diameter, hose velocity, and nozzle velocity, we can solve for the nozzle's diameter. This involves algebraic manipulation and possibly the use of a calculator to find the precise value of the nozzle's diameter in centimeters.