A landscape architect is planing an artificial water fall. Water flowing at 1.11 m/s will leave the end of a horizontal channel at the top of a vertical wall h=4.00 m high. Will the space behind the waterfall be wide enough for a pedistrian walkway? To sell her plan to the city council, the architect wants to build a model to scale which is 1/14 actual size. How fast should the water flow in the channel in the model?
To determine whether the space behind the waterfall will be wide enough for a pedestrian walkway, we need to calculate the range of the water flow at the bottom.
Given:
Initial water speed, v1 = 1.11 m/s
Height of the waterfall, h = 4.00 m
The range of the water flow can be determined using the range formula for projectile motion:
R = (v1^2/g) * sin(2θ)
Since the water flows horizontally off a vertical wall, the initial angle of projection (θ) is 90 degrees.
The range formula simplifies to:
R = (v1^2/g)
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Now let's calculate the actual range of the water flow:
R = (1.11^2/9.8) m
R ≈ 0.1254 m
Since the scale of the model is 1/14, the scaled range in the model can be found by dividing the actual range by 14:
Scaled range = (0.1254 m) / 14
Scaled range ≈ 0.00896 m
Therefore, the water flow in the channel of the model should be approximately 0.00896 m/s in order to maintain the scale.
To determine if the space behind the waterfall will be wide enough for a pedestrian walkway, we first need to calculate the range of the water.
Given:
Water flow velocity (V) = 1.11 m/s
Height of the waterfall (h) = 4.00 m
To calculate the range of the water, we can use the formula:
Range = V * sqrt(2h/g)
Where:
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Plugging in the values, we have:
Range = 1.11 * sqrt(2 * 4.00 / 9.8) = 1.11 * sqrt(0.8163) = 1.11 * 0.9032 = 1.0022 m
So the range of the water will be approximately 1.00 meters.
Now let's determine the necessary width of the space behind the waterfall for a pedestrian walkway.
Typically, a comfortable minimum width for a pedestrian walkway is around 1.20 meters.
Since the range of the water is approximately 1.00 meters, the necessary width for the space behind the waterfall would be slightly more than 1.20 meters, to allow for the water's range, as well as a safe walking distance.
As for the second part of the question, since the architect wants to build a model that is 1/14 actual size, we need to calculate the corresponding velocity for the water in the model.
The velocity in the model should be scaled down by the same factor as the dimensions, so:
Velocity in the model = Velocity in actual size / Scale factor
The scale factor is 1/14, so:
Velocity in the model = 1.11 m/s / 14 = 0.0793 m/s
Therefore, the water should flow at a velocity of approximately 0.0793 m/s in the channel of the scaled-down model.