Question

The Huka Falls on the Waikato River is one of New Zealand’s most visited natural tourist attractions (see Figure 12.29). On average the river has a flow rate of about 300,000 L/s. At the gorge, the river narrows to 19 m wide and averages 20 m deep. (a) What is the average speed of the river in the gorge? (b) What is the average speed of the water in the river downstream of the falls when it widens to 60 m and its depth increases to an average of 40 m?

Answer 1

To find the average speed of the river in the gorge and downstream of the falls, we can use the principle of conservation of mass, which states that the mass flow rate of a fluid remains constant.

Given:

Flow rate of the river = 300,000 L/s

Width of the gorge = 19 m

Depth in the gorge = 20 m

Width downstream of the falls = 60 m

Depth downstream of the falls = 40 m

(a) Average speed of the river in the gorge:

First, we need to calculate the cross-sectional area of the river in the gorge:

Area = Width × Depth

Area = 19 m × 20 m

Area = 380 m^2

Now, we can calculate the average speed in the gorge:

Average speed = Flow rate / Area

Average speed = 300,000 L/s / 380 m^2

Average speed ≈ 789.47 m/s

(b) Average speed of the water downstream of the falls:

Similarly, we calculate the cross-sectional area downstream of the falls:

Area = Width × Depth

Area = 60 m × 40 m

Area = 2400 m^2

Now, we can calculate the average speed downstream of the falls:

Average speed = Flow rate / Area

Average speed = 300,000 L/s / 2400 m^2

Average speed ≈ 125 m/s

Therefore, the average speed of the river in the gorge is approximately 789.47 m/s, and the average speed of the water downstream of the falls is approximately 125 m/s.