A ball is dropped from a height of 20m above the surface of water in a lake. The refractive index of water is `4//3`. A fish inside the lake , in the line fall of the ball, is looking at the ball. At an instant when the balll is 12.8m above the water surface, the fish sees the speed of ball as .

Options

A. `9ms^(-1)`
B. `12ms^(-1)`
C. `16ms^(-1)`
D. `21.33ms^(-1)`

Explanation

To solve the problem, we need to find the speed of the ball as seen by the fish in the water when the ball is at a height of 12.8 meters above the water surface. We'll follow these steps:

### Step 1: Calculate the height the ball has fallen
The ball is dropped from a height of 20 meters, and at the instant we are considering, it is 12.8 meters above the water surface.

\[ \text{Height fallen} = \text{Initial height} - \text{Current height} = 20 \, \text{m} - 12.8 \, \text{m} = 7.2 \, \text{m} \]

### Step 2: Calculate the speed of the ball just before it reaches 12.8 meters
We can use the third equation of motion to find the speed of the ball after falling 7.2 meters. The equation is:

\[ v^2 = u^2 + 2as \]

Where:
- \(v\) = final velocity
- \(u\) = initial velocity (0 m/s, since the ball is dropped)
- \(a\) = acceleration due to gravity (approximately \(10 \, \text{m/s}^2\))
- \(s\) = distance fallen (7.2 m)

Substituting the values:

\[ v^2 = 0 + 2 \cdot 10 \cdot 7.2 \]
\[ v^2 = 144 \]
\[ v = \sqrt{144} = 12 \, \text{m/s} \]

### Step 3: Calculate the apparent speed of the ball as seen by the fish
The speed of the ball as seen by the fish in the water can be calculated using the refractive index. The relationship between the real speed (\(v_{real}\)) and the apparent speed (\(v_{apparent}\)) is given by:

\[ v_{apparent} = \mu \cdot v_{real} \]

Where:
- \(\mu\) = refractive index of water = \(\frac{4}{3}\)
- \(v_{real} = 12 \, \text{m/s}\)

Substituting the values:

\[ v_{apparent} = \frac{4}{3} \cdot 12 \]
\[ v_{apparent} = \frac{48}{3} = 16 \, \text{m/s} \]

### Final Answer
The speed of the ball as seen by the fish is \(16 \, \text{m/s}\).

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