A fruit which is at 20 m above a lake suddenly detaches from the tree.A fish inside the lake , in the line of fall of fruit,is looking at fruit when fruit is at 12.8 m above the water surface.Speed of fruit as observed by fish `(μ_(water)=4/3 and g=10m/s^2)`

Options

A. 6 m/s
B. 12 m/s
C. 15 m/s
D. 16 m/s

Answer

To solve the problem of determining the speed of the fruit as observed by the fish, we will follow these steps:

### Step 1: Determine the height of the fruit above the lake
The fruit is initially at a height of 20 m above the lake. When the fish observes the fruit, it is at a height of 12.8 m above the water surface.

### Step 2: Calculate the distance fallen by the fruit
The distance fallen by the fruit can be calculated as:
\[ \text{Distance fallen} = \text{Initial height} - \text{Current height} = 20 \, \text{m} - 12.8 \, \text{m} = 7.2 \, \text{m} \]

### Step 3: Calculate the time taken to fall this distance
Using the equation of motion for free fall:
\[ s = ut + \frac{1}{2}gt^2 \]
where \(s\) is the distance fallen (7.2 m), \(u\) is the initial velocity (0 m/s, since it starts from rest), \(g\) is the acceleration due to gravity (10 m/s²), and \(t\) is the time in seconds.

Substituting the values:
\[ 7.2 = 0 + \frac{1}{2} \cdot 10 \cdot t^2 \]
\[ 7.2 = 5t^2 \]
\[ t^2 = \frac{7.2}{5} = 1.44 \]
\[ t = \sqrt{1.44} = 1.2 \, \text{s} \]

### Step 4: Calculate the speed of the fruit just before it is observed by the fish
The speed of the fruit just before it is observed can be calculated using the formula:
\[ v = u + gt \]
Substituting the known values:
\[ v = 0 + 10 \cdot 1.2 = 12 \, \text{m/s} \]

### Step 5: Apply the refractive index to find the speed observed by the fish
Since the fish is in water (denser medium) and the fruit is in air (rarer medium), the speed of the fruit as observed by the fish is given by:
\[ v' = \mu \cdot v \]
where \( \mu = \frac{4}{3} \) (the refractive index of water) and \( v = 12 \, \text{m/s} \).

Substituting the values:
\[ v' = \frac{4}{3} \cdot 12 = 16 \, \text{m/s} \]

### Final Answer
The speed of the fruit as observed by the fish is **16 m/s**.

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