Answer:
The average velocity of the stream with a depth of 10 feet is greater.
(a) is correct option.
Explanation:
Given that,
Depth [tex]h_{1}= 10\ feet[/tex]
Depth [tex]h_{2}=6\ feet[/tex]
We need to calculate the average velocity of the stream
According to question,
[tex]h_{1} > h_{2}[/tex]
The velocity for first case,
[tex]v_{1}=u_{1}\dfrac{x_{1}}{h_{1}}[/tex]
[tex]\dfrac{v_{1}}{x_{1}}=\dfrac{u_{1}}{h_{1}}[/tex]
The velocity for second case,
[tex]v_{2}=u_{2}\dfrac{x_{2}}{h_{2}}[/tex]
[tex]\dfrac{v_{2}}{x_{2}}=\dfrac{u_{2}}{h_{2}}[/tex]
For the same velocity profile,
[tex]\dfrac{dv}{dx}=\dfrac{v_{1}}{x_{1}}=\dfrac{v_{2}}{x_{2}}[/tex]
Then,
[tex]\dfrac{u_{1}}{h_{1}}=\dfrac{u_{2}}{h_{2}}[/tex]
Put the value into the formula
[tex]\dfrac{u_{1}}{10}=\dfrac{u_{2}}{6}[/tex]
[tex]u_{1}=\dfrac{5}{3}u_{2}[/tex]
[tex]u_{1}=1.67u_{2}[/tex]
The velocity is [tex]u_{1} > u_{2}[/tex]
We need to calculate the average velocity for first case
Using formula of average velocity
[tex]v_{avg}_{1}=\dfrac{0+u_{1}}{2}[/tex]
Put the value into the formula
[tex]v_{avg}_{1}=\dfrac{0+u_{1}}{2}[/tex]
[tex]v_{avg}_{1}=\dfrac{u_{1}}{2}[/tex]
We need to calculate the average velocity for second case
Using formula of average velocity
[tex]v_{avg}_{2}=\dfrac{0+u_{2}}{2}[/tex]
Put the value into the formula
[tex]v_{avg}_{2}=\dfrac{0+u_{2}}{2}[/tex]
[tex]v_{avg}_{2}=\dfrac{u_{2}}{2}[/tex]
If [tex]u_{1} > u_{2}[/tex] then [tex]\dfrac{u_{1}}{2} >\dfrac{u_{2}}{2}[/tex]
So, we can say that the average velocity of the stream with a depth of 10 feet will be greater than the stream with a depth of 6 feet.
Hence, The average velocity of the stream with a depth of 10 feet is greater.
(a) is correct option.
