The Huka falls on the Waikato River is one of New Zealand’s most visited natural tourist attractions. On average, the river has a flow rate of about 300000 liters per second. At the gorge, the river narrows to 20-meters wide and averages 20-meters deep. What is the average speed of the river in the gorge? What is the average speed of the water in the river downstream of the falls when it widens to 60 meters and its depth increases to an average of 40 meters?
We’ll call the flow rate of the water in the river, 300000 liters per second, 𝑅. And we’ll call the average speed of the river in the gorge 𝑣 sub 𝑔. And we’ll name the speed of the water in the river downstream of the falls 𝑣 sub 𝑑.
We’ll begin by solving for 𝑣 sub 𝑔, the average speed of the river in the gorge. If we look at the cross section of the river at our two locations, at the gorge and downstream of the gorge, we’re told the average dimensions of the river’s width and depth at these two points. At the gorge, the river on average is 20-meters deep and 20-meters wide. And farther downstream, on average, it’s 60-meters wide and 40-meters deep. With a total flow rate of 𝑅 equals 300000 liters per second, we wanna know how fast the water goes through the gorge, and how fast it goes further downstream.
To solve for the speed of the water through the gorge, we can divide the overall flow rate by the cross-sectional area of the river. Here, capital 𝑅 is our flow rate and 𝐴 sub 𝑔 is the cross-sectional area of the river at the gorge. If we replace 𝐴 sub 𝑔 with the product of the average width and average depth of the river at the gorge, then we see our area will be in units of meters squared. However, our given rate 𝑅 is in units of liters per second. So to convert liters into cubic meters, let’s recall the volume conversion relationship between those two units. 1000 liters is equal to one meter cubed, so 𝑅, the flow rate through the river of 300000 liters per second, is equal to 300 cubic meters per second. When we multiply the denominator of this fraction, we find 400 meters squared.
Notice that when we perform this division, the units leftover will be meters per second, a speed. 300 cubic meters per second divided by 400 meters squared equals 0.75 meters per second. That’s the average speed of the river as it passes through the gorge.
Now let’s solve for the average speed of the water as it moves through the river downstream of the gorge, 𝑣 sub 𝑑. That’s equal to the same flow rate divided by the cross-sectional area of the river downstream, 𝑎 sub 𝑑. The flow rate of water through the river is always constant, so 𝑅 is, again, 300 meters cubed per second and 𝐴 sub 𝑑 is equal to 60 meters multiplied by 40 meters which, multiplied together, equals 2400 square meters.
When we perform this division and round to two significant figures, the average speed of the water downstream of the gorge is equal to 0.13 meters per second. So as the river widens out and becomes deeper, the flow rate of the water decreases.