Final answer:
The student's question involves calculating the flow rate and nozzle velocity for water flowing through a hose using the equation of continuity and fluid dynamics principles. The flow rate is found with the equation Q = A * v₁, and the nozzle's velocity is determined using the conservation of mass.
Step-by-step explanation:
Calculating the Velocity of Water in a Hose's Nozzle
The question relates to the application of fluid dynamics principles, specifically the equation of continuity and flow rate, to determine the velocity of water in a hose's nozzle. Given the velocity of water through a hose (v₁=2.0 m/s) and its internal diameter (1.6 cm), we are asked to calculate: (a) the flow rate; and (b) the velocity of water in the nozzle of the hose (v₂) based on a new given velocity of 15.0 m/s.
Flow Rate Calculation (a)
We start with the flow rate equation Q = A * v₁, where Q is the flow rate and A is the cross-sectional area. The area A of a hose is given by π * (r₁)². After calculating the area using the given diameter, we find the flow rate in cubic meters per second and convert it to liters per second since 1 m³ equals 1000 liters.
Velocity in the Nozzle (b)
To find the nozzle's velocity v₂, we apply the equation of continuity A₁ * v₁ = A₂ * v₂. With A₁ calculated from the hose diameter, the new velocity at the nozzle (v₂=15.0 m/s), we can rearrange this equation to solve for the cross-sectional area of the nozzle A₂, and from it, calculate the diameter.
In summary, we utilize the known velocities and diameters along with the principle of conservation of mass to determine the flow rate and nozzle velocity. Remember, to find the nozzle's inside diameter from its area, one would need to solve for radius (r=√(A/π)) and then double it to get the diameter.