Problem Analysis:

• Given: Flow rate ($Q$) = 15 m$^3$/s, Channel width ($B$) = 5 m, Gravity ($g$) = 9.81 m/s$^2$.
• Objective: Calculate the critical velocity ($V_c$) for the flow.
• The correct answer lies between 3.025 m/s and 3.125 m/s.

Calculating Critical Velocity

For a rectangular channel, the critical velocity ($V_c$) can be calculated using the formula derived from the Froude number condition ($Fr = 1$). The formula relates critical velocity directly to the flow rate per unit width ($q$) and gravity ($g$):

$V_c = \sqrt[3]{g q}$

Where:

• $q$ is the flow rate per unit width.
• $q = \frac{Q}{B}

Step 1: Calculate Flow Rate per Unit Width ($q$)

First, find the flow rate per unit width ($q$) by dividing the total flow rate ($Q$) by the channel width ($B$):

$q = \frac{15 \text{ m}^3/\text{s}}{5 \text{ m}} = 3 \text{ m}^2/\text{s}$

Step 2: Calculate Critical Velocity ($V_c$)

Now, substitute the values of $g$ and $q$ into the critical velocity formula:

$V_c = \sqrt[3]{g q}$

$V_c = \sqrt[3]{(9.81 \text{ m/s}^2) \times (3 \text{ m}^2/\text{s})}$

$V_c = \sqrt[3]{29.43 \text{ m}^3/\text{s}^3}$

$V_c \approx 3.0885 \text{ m/s}$

Step 3: Rounding the Result

Rounding the calculated critical velocity to three decimal places:

$V_c \approx 3.089 \text{ m/s}$

This value falls within the specified range of 3.025 to 3.125 m/s.