Question

You are working as an assistant to a landscape architect, who is planning an artificial waterfall in a city park. Water flowing at 1.93 m/s will leave the end of a horizontal channel at the top of a vertical wall h = 3.05 m high, and from there it will fall into a pool (see the figure below) (a) The architect has asked you to determine if the space behind the waterfall will be wide enough for a pedestrian walkway. (Enter the maximum width, in meters, of a walkway behind the waterfall.) (b) To sell her plan to the city council, the architect wants to build a model to standard scale, which is one-twelfth actual size. She asks you to design and build the model, with the important question being: how fast (in m/s) should the water flow in the channel in the model? m/s

Answer 1

The maximum width of the walkway behind the waterfall can be 1.52 meters. For the model, which is one-twelfth the actual size, the water should flow at approximately 0.56 m/s to maintain dynamic similarity with the full-scale waterfall.

Step-by-step explanation:

This physics problem requires an understanding of projectile motion principles and the application of Bernoulli's principle for fluid flow in the design and scale modeling of an artificial waterfall. We will calculate the distance the water travels horizontally before hitting the ground to determine the walkway width, and use scaling laws to find the model water flow speed.

Walkway Width Calculation

Given the horizontal velocity (1.93 m/s) and the height (3.05 m), we can use the kinematic equation Δy = Vyi Δt + 0.5gΔt², where Vyi = 0 since there is no initial vertical velocity component, to calculate the time it takes for the water to fall:

Δt = √(2Δy / g)
Δt = √(2*3.05 m / 9.81 m/s²)
Δt ≈ 0.79 s

The horizontal distance traveled (maximum walkway width) can be found by Δx = Vxi Δt:

Δx = 1.93 m/s * 0.79 s
Δx ≈ 1.52 m

Model Water Flow Speed Calculation

For the scale model 1/12th the size, the flow rate should be the same as in the full-scale model, which leads to the velocity in the model being proportionally less due to the smaller cross-sectional area. To maintain dynamic similarity, the velocity should be scaled by the square root of the linear scale factor (since velocity is related to the square root of scale in fluid dynamics), which means:

V_model = V_actual √(scale factor)
V_model = 1.93 m/s √(1/12)
V_model ≈ 0.56 m/s