At Deoprayag (Garhwal, UP) river Alaknanda mixes with the river Bhagirathi and becomes river Ganga. Suppose Alaknanda has a width of 12 m, Bhagirathi has a width of 8m and Ganga has a width of 16m. Assume that the depth of water is same in the three rivers. Let the average speed of water in Alaknanda be `20 kmh^-1` and in Bhagirathi be `16 kmh^-1`. Find the average speed of water in the river Ganga.

Explanation

To find the average speed of water in the river Ganga after the confluence of the rivers Alaknanda and Bhagirathi, we can use the principle of conservation of mass, which in fluid mechanics is often expressed through the continuity equation. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have three rivers: Alaknanda, Bhagirathi, and Ganga. We know their widths and the average speeds of water in Alaknanda and Bhagirathi. We need to find the average speed of water in Ganga.

### Step 2: Gather Given Data
- Width of Alaknanda, \( B_A = 12 \, \text{m} \)
- Speed of Alaknanda, \( V_A = 20 \, \text{km/h} \)
- Width of Bhagirathi, \( B_B = 8 \, \text{m} \)
- Speed of Bhagirathi, \( V_B = 16 \, \text{km/h} \)
- Width of Ganga, \( B_G = 16 \, \text{m} \)
- Depth of water is the same in all rivers (denote as \( D \))

### Step 3: Apply the Continuity Equation
According to the continuity equation, the mass flow rate entering the junction (where the rivers meet) must equal the mass flow rate exiting the junction.

The mass flow rate can be expressed as:
\[ \text{Mass Flow Rate} = \text{Density} \times \text{Area} \times \text{Velocity} \]

Since the density of water is constant, we can simplify our equation by focusing on the areas and velocities:
\[ B_A \cdot V_A + B_B \cdot V_B = B_G \cdot V_G \]

### Step 4: Substitute the Known Values
Substituting the known values into the equation:
\[ 12 \cdot 20 + 8 \cdot 16 = 16 \cdot V_G \]

### Step 5: Calculate the Left Side
Calculating the left side:
\[ 12 \cdot 20 = 240 \quad \text{(for Alaknanda)} \]
\[ 8 \cdot 16 = 128 \quad \text{(for Bhagirathi)} \]
Adding these together:
\[ 240 + 128 = 368 \]

### Step 6: Set Up the Equation for Ganga
Now we have:
\[ 368 = 16 \cdot V_G \]

### Step 7: Solve for \( V_G \)
To find \( V_G \), divide both sides by 16:
\[ V_G = \frac{368}{16} \]
Calculating this gives:
\[ V_G = 23 \, \text{km/h} \]

### Final Answer
The average speed of water in the river Ganga is \( 23 \, \text{km/h} \).

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