Step-by-step explanation:
To find the speed of water through each of the smaller pipes, we can use the principle of continuity, which states that the volume flow rate of fluid remains constant as it flows through a pipe.
The volume flow rate (Q) is given by: Q = A * V, where A is the cross-sectional area of the pipe and V is the velocity of the water.
For the larger pipe:
Diameter (D) = 50.0 cm = 0.50 m
Radius (r1) of the larger pipe = D/2 = 0.50 m / 2 = 0.25 m
Cross-sectional area (A1) of the larger pipe = π * r1^2 = 3.1416 * (0.25)^2 ≈ 0.19635 m²
Velocity (V1) of the water in the larger pipe = 25.0 m/s
Volume flow rate (Q1) through the larger pipe = A1 * V1 ≈ 0.19635 m² * 25.0 m/s ≈ 4.90875 m³/s
Now, since the volume flow rate is constant, it remains the same as water branches into fifty identical smaller pipes.
For each of the smaller pipes:
Diameter (D) = 4.00 cm = 0.04 m
Radius (r2) of each smaller pipe = D/2 = 0.04 m / 2 = 0.02 m
Cross-sectional area (A2) of each smaller pipe = π * r2^2 = 3.1416 * (0.02)^2 ≈ 0.00125664 m²
Now we can find the velocity (V2) of the water in each smaller pipe using the volume flow rate (Q1) and the cross-sectional area (A2) of each smaller pipe:
Q1 = A2 * V2
V2 = Q1 / A2 ≈ 4.90875 m³/s / 0.00125664 m² ≈ 3906.25 m/s
So, the speed of the water through each of the smaller pipes is approximately 3906.25 m/s. However, please note that this result seems unusually high, and it's possible there might be an error in the input values or the calculation. Please double-check the values and the problem statement to ensure accuracy.