Question

A river 16.0 m wide and 4.0 m deep drains a 3000.0 km2 land area in which the average precipitation is 48 cm/year. One-fourth of this rainfall returns to the atmosphere by evaporation, but the remainder ultimately drains into the river. What is the average speed of the river?

Answer 1

This question requires knowledge of flow rates Q, which is defined as the product of the velocity v of a liquid times the area A through which it passes:

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Step 1) Since the the river has to carry away all the water that the land received in the form of rain (minus that which returned to the atmosphere), the flow rates of the water in the river must match that of the water hitting the wide area in the form of rain:

The v₁ is the velocity of the river, A₁ is the cross-sectional area of the river, v₂ is the velocity of the rain, and A₂ is the area over which the rain lands.

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Step 2) Solve for v₁:

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Step 3) Use that since 1/4 of the rain ultimately returns to the atmosphere, that means that the equivalent amount of rainfall is 36 cm/year instead of 48 cm/year.

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Step 4) Rewrite that velocity with standard units of meters and seconds:

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Step 5) Convert the 3,000 km² into standard units:

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Step 6) Find the cross-sectional area of the river, using the dimensions of 16.0 m and 4.0 m:

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Step 7) Use 1.14*10^-8 m/s for the velocity of the rain v₂, 3*10⁹ m² for the area of the land A₂, and 64.0 m² for the cross-sectional area of the river A₁ in the equation from step 2:

So the river should have a velocity of about 0.5 m/s.