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?

b. 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 if the space behind the waterfall is wide enough for a pedestrian walkway, we need to consider the horizontal distance the water will travel when it falls from a height of 4.00 m.

To calculate this, we'll use the principle of projectile motion. The water leaving the end of the horizontal channel can be considered a projectile.

Using the kinematic equation for vertical motion:

v_f² = v_i² + 2*a*d

Where:
v_f = final velocity (0 m/s at the highest point of the projectile motion)
v_i = initial velocity (1.11 m/s)
a = acceleration due to gravity (approximately -9.81 m/s², accounting for the downward direction)
d = vertical displacement (4.00 m)

Plugging in the values:

0 = (1.11 m/s)² + 2*(-9.81 m/s²)*4.00 m

Simplifying:

0 = 1.2321 m²/s² - 78.48 m²/s²

Rearranging the equation to solve for the vertical displacement:

d = (1.2321 m²/s²)/(2*(-9.81 m/s²))
d = 0.0629 m

Thus, the water will fall a vertical distance of approximately 0.0629 meters.

Now, let's calculate the horizontal distance the water will travel. Since the channel is horizontal, there is no horizontal acceleration, so the initial horizontal velocity will remain constant throughout the motion.

The horizontal distance (x) can be calculated using the formula:

x = v_i * t

Where:
v_i = initial velocity (1.11 m/s)
t = time of flight

To find the time of flight, we can use the equation for vertical motion:

d = v_i * t + 0.5 * a * t²

Substituting the values:

0.0629 m = 1.11 m/s * t + 0.5 * (-9.81 m/s²) * t²

Simplifying:

-4.905 t² + 1.11 t + 0.0629 = 0

Solving this quadratic equation for t will give us the time of flight. Using the quadratic formula:

t = (-1.11 ± √((1.11)² - 4*(-4.905)*(0.0629)))/(2*(-4.905))

Calculating t using a positive sign in the ±:

t = (-1.11 + √((1.11)² - 4*(-4.905)*(0.0629)))/(2*(-4.905))
t ≈ 0.258 s

Now, let's calculate the horizontal distance (x):

x = 1.11 m/s * 0.258 s
x ≈ 0.286 m

Therefore, the water will travel approximately 0.286 meters horizontally.

To determine if the space behind the waterfall is wide enough for a pedestrian walkway, compare this horizontal distance to the desired width of the walkway.

Now, let's address the second part of the question regarding the scale model.

Since the scale of the model is 1/14 of the actual size, the velocities in the model will be scaled accordingly. The speed of the water flowing in the channel in the model should also be reduced by a factor of 1/14.

Therefore, the water should flow in the channel in the model at a speed of:

1.11 m/s * (1/14)
≈ 0.079 m/s

Hence, the water in the model should flow at approximately 0.079 m/s.