A fish is rising up vertically inside a pond with velocity of `4cm//s`, and notices a bird , which is diving vertically downward and its velocity appears to be `16cm//s` (to th fish). What is the real velocity of the diving bird, if refractive index of water is `4//3` ?

Answer

To solve the problem, we need to find the real velocity of the diving bird as observed by the fish, given the apparent velocity and the refractive index of water. Here’s a step-by-step solution:

### Step 1: Understand the given information
- The fish is rising vertically with a velocity \( v_f = 4 \, \text{cm/s} \).
- The apparent velocity of the bird as seen by the fish is \( v_a = 16 \, \text{cm/s} \).
- The refractive index of water is \( n = \frac{4}{3} \).

### Step 2: Relate the velocities using the refractive index
The relationship between the apparent velocity \( v_a \), the real velocity of the bird \( v_b \), and the velocity of the fish \( v_f \) can be expressed as:
\[ v_a = v_b + n \cdot v_f \]
Where:
- \( v_a \) is the apparent velocity of the bird,
- \( v_b \) is the real velocity of the bird,
- \( n \) is the refractive index of water,
- \( v_f \) is the velocity of the fish.

### Step 3: Substitute the known values into the equation
Substituting the known values into the equation:
\[ 16 = v_b + \left(\frac{4}{3}\right) \cdot 4 \]

### Step 4: Calculate the term involving the refractive index
Calculate \( n \cdot v_f \):
\[ n \cdot v_f = \frac{4}{3} \cdot 4 = \frac{16}{3} \, \text{cm/s} \]

### Step 5: Substitute this value back into the equation
Now, substitute this back into the equation:
\[ 16 = v_b + \frac{16}{3} \]

### Step 6: Solve for the real velocity of the bird \( v_b \)
To isolate \( v_b \), we rearrange the equation:
\[ v_b = 16 - \frac{16}{3} \]
Convert 16 into a fraction with a denominator of 3:
\[ 16 = \frac{48}{3} \]
Now substitute:
\[ v_b = \frac{48}{3} - \frac{16}{3} = \frac{32}{3} \, \text{cm/s} \]

### Step 7: Convert to decimal form (if necessary)
Calculating \( \frac{32}{3} \):
\[ v_b \approx 10.67 \, \text{cm/s} \]

### Final Answer
The real velocity of the diving bird is approximately \( 10.67 \, \text{cm/s} \).

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