If the average discharge of the river at reach AB is 200 m³/s and increases to 300 m³/s at reach CD, then the stream power from the reach AB to CD will change by a factor of ______... If the average discharge of the river at reach AB is 200 m³/s and increases to 300 m³/s at reach CD, then the stream power from the reach AB to CD will change by a factor of __________. (Round off to two decimal places)
Understand the Problem
The question is asking for the change in stream power from reach AB to CD in the river, given the discharge values at these points. The approach will involve calculating stream power using the formula that relates flow rate, gravitational acceleration, and water density.
Answer
The stream power changes by a factor of $1.50$.
Answer for screen readers
The stream power from reach AB to CD will change by a factor of $1.50$.
Steps to Solve
- Understanding Stream Power Formula
Stream power ($S$) can be calculated using the formula: $$ S = Q \cdot \rho \cdot g $$ where
- $Q$ is the discharge (in m³/s),
- $\rho$ is the water density (in kg/m³),
- $g$ is the gravitational acceleration (in m/s²).
- Calculate Stream Power at Reach AB
Substituting the values for reach AB where $Q_{AB} = 200 , \text{m³/s}$: $$ S_{AB} = 200 \cdot 1000 \cdot 10 $$
- Calculate Stream Power at Reach CD
For reach CD where $Q_{CD} = 300 , \text{m³/s}$: $$ S_{CD} = 300 \cdot 1000 \cdot 10 $$
- Calculating Stream Power Values
Calculate both values:
-
For $S_{AB}$: $$ S_{AB} = 2000000 , \text{W} $$
-
For $S_{CD}$: $$ S_{CD} = 3000000 , \text{W} $$
- Finding the Change Factor in Stream Power
To find the factor by which stream power changes from reach AB to CD: $$ \text{Factor} = \frac{S_{CD}}{S_{AB}} = \frac{3000000}{2000000} $$
- Final Calculation
Compute: $$ \text{Factor} = 1.5 $$
The stream power from reach AB to CD will change by a factor of $1.50$.
More Information
The stream power influences river dynamics and sediment transport, showing how discharge variations affect the river's ability to do work.
Tips
- Mixing up the formulas for power and other quantities can occur; ensure you're using the correct formula for stream power specifically.
- Failing to convert units correctly or not keeping them consistent can lead to errors in the final answer.
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