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Chapter 14: Problem 14

Two streams merge to form a river. One stream has a width of \(8.2 \mathrm{~m}\), depth of \(3.4 \mathrm{~m}\), and current speed of \(2.3 \mathrm{~m} / \mathrm{s}\). The other stream is \(6.8 \mathrm{~m}\) wide and \(3.2 \mathrm{~m}\) deep, and flows at \(2.6 \mathrm{~m} / \mathrm{s}\). If the river has width \(10.5 \mathrm{~m}\) and depth \(4.5 \mathrm{~m}\), what is its speed?

Short Answer

Expert verified
The speed of the river is approximately 2.55 m/s.

Step by step solution

01

Calculate the flow rate of the first stream

The flow rate can be calculated using this formula: \( Q = ext{width} \times ext{depth} \times ext{speed} \).For the first stream:- Width = 8.2 m- Depth = 3.4 m- Speed = 2.3 m/sSo, \( Q_1 = 8.2 \times 3.4 \times 2.3 = 64.052 \) cubic meters per second.
02

Calculate the flow rate of the second stream

Using the same formula, we calculate the flow rate for the second stream:- Width = 6.8 m- Depth = 3.2 m- Speed = 2.6 m/sSo, \( Q_2 = 6.8 \times 3.2 \times 2.6 = 56.576 \) cubic meters per second.
03

Find the total flow rate of the river

The total flow rate of the river is the sum of the flow rates of the two streams:\( Q_{total} = Q_1 + Q_2 = 64.052 + 56.576 = 120.628 \) cubic meters per second.
04

Calculate the speed of the river

The speed of the river can be calculated using the flow rate equation rearranged as:\( v = \frac{Q_{total}}{ ext{width} \times ext{depth}} \).For the river:- Total flow rate \( Q_{total} = 120.628 \) cubic meters per second- Width = 10.5 m- Depth = 4.5 mTherefore, the speed of the river is:\( v = \frac{120.628}{10.5 \times 4.5} = \frac{120.628}{47.25} \approx 2.55 \) meters per second.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Flow Rate Calculation
Understanding flow rate is essential in fluid mechanics as it quantifies the volume of fluid passing a point per unit time. To calculate the flow rate (\( Q )\), the formula used is:
\[ Q = \text{width} \times \text{depth} \times \text{speed} \]This is the product of the cross-sectional area of the flow (width times depth) and the speed of the fluid. In practical terms, this calculation tells us how much water a stream or river carries over a period.
  • Width: This is the horizontal span of the water body.
  • Depth: This measures how deep the water is, on average.
  • Speed: This is the velocity at which the water is flowing.
Using the details of the streams merging into the river, we calculated the flow rate for each using the above formula. Knowing the flow rates will help us determine how much water is contributed by each stream.
River Velocity
The velocity of water in a river is crucial for predicting the river's behavior and managing its flow. Once the total flow rate of all contributing streams is known, the velocity of the river can be calculated. This involves rearranging the flow rate formula to solve for velocity:
\[ v = \frac{Q_{\text{total}}}{\text{width} \times \text{depth}} \]It's important to understand that a river鈥檚 velocity will affect not only its ability to erode banks and transport sediment but also the safety and navigability of the waterway.
  • The total flow rate is the combined flow rates of the contributing bodies of water.
  • The river width and depth are where these flows merge.
In our problem, after calculating the total flow rate of the river from the two streams, we used this formula to find the river's speed was approximately 2.55 meters per second.
Continuity Equation
The continuity equation is a fundamental principle in fluid mechanics that expresses the conservation of mass in a fluid flow. Essentially, it ensures what goes into a flow system must come out. The basic essence of the continuity equation is:
\[ A_1 \cdot v_1 = A_2 \cdot v_2 = Q \]Where:
  • A represents the cross-sectional area at points 1 and 2.
  • v represents the fluid's velocity at these points.
  • Q is the flow rate, which must remain constant if no fluid is added or removed.
In our scenario with merging streams forming a river, the continuity principle ensures the total incoming flow from the streams equals the river's outgoing flow. This equation helps us calculate unknown values when two of the three variables are known. It鈥檚 particularly useful when dealing with situations involving varying cross-sections, like rivers or pipes.

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Most popular questions from this chapter

What fraction of the volume of an iceberg (density \(917 \mathrm{~kg} / \mathrm{m}^{3}\) ) would be visible if the iceberg floats (a) in the ocean (salt water, density \(1024 \mathrm{~kg} / \mathrm{m}^{3}\) ) and (b) in a river (fresh water, density \(1000 \mathrm{~kg} / \mathrm{m}^{3}\) )? (When salt water freezes to form ice, the salt is excluded. So, an iceberg could provide fresh water to a community.)

Suppose that you release a small ball from rest at a depth of \(0.400 \mathrm{~m}\) below the surface in a pool of water. If the density of the ball is \(0.450\) that of water and if the drag force on the ball from the water is negligible, how high above the water surface will the ball shoot as it emerges from the water? (Neglect any transfer of energy to the splashing and waves produced by the emerging ball.)

Water is moving with a speed of \(5.0 \mathrm{~m} / \mathrm{s}\) through a pipe with a cross-sectional area of \(4.0 \mathrm{~cm}^{2}\). The water gradually descends \(12 \mathrm{~m}\) as the pipe cross-sectional area increases to \(8.0 \mathrm{~cm}^{2}\). (a) What is the speed at the lower level? (b) If the pressure at the upper level is \(1.5 \times 10^{5} \mathrm{~Pa}\), what is the pressure at the lower level?

In 1654 Otto von Guericke, inventor of the air pump, gave a demonstration before the noblemen of the Holy Roman Empire in which two teams of eight horses could not pull apart two evacuated brass hemispheres. (a) Assuming the hemispheres have (strong) thin walls, so that \(R\) in Fig. 14-44 may be considered both the inside and outside radius, show that the force \(\vec{F}\) required to pull apart the hemispheres has magnitude \(F=\pi R^{2} \Delta p\), where \(\Delta p\) is the difference between the pressures outside and inside the sphere. (b) Taking \(R\) as \(40 \mathrm{~cm}\), the inside pressure as \(0.10 \mathrm{~atm}\), and the outside pressure as \(1.00 \mathrm{~atm}\), find the force magnitude the teams of horses would have had to exert to pull apart the hemispheres. (c) Explain why one team of horses could have proved the point just as well if the hemispheres were attached to a sturdy wall.

Two identical cylindrical vessels with their bases at the same level each contain a liquid of density \(1.30 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\). The area of each base is \(4.25 \mathrm{~cm}^{2}\), but in one vessel the liquid height is \(0.854 \mathrm{~m}\) and in the other it is \(1.560 \mathrm{~m}\). Find the work done by the gravitational force in equalizing the levels when the two vessels are connected.
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