A lead ball dropped into a lake from a diving board 5 m above the water hits the water with certain velocity and then sinks to the bottom with the same constant velocity. If it reaches the bottom in 3s after it is dropped the depth of the lake is `(g = 10 ms^(-2))`

Options

A. 30 m
B. 15 m
C. 10 m
D. 20 m

Answer

To solve the problem step by step, we will break down the motion of the lead ball into two parts: the free fall from the diving board to the water surface and the sinking from the water surface to the bottom of the lake.

### Step 1: Analyze the motion from the diving board to the water surface
- The ball is dropped from a height of 5 meters (AO = 5 m).
- Initial velocity (u) = 0 m/s (since it is dropped).
- We need to find the final velocity (v) just before it hits the water.

Using the equation of motion:
\[ v^2 = u^2 + 2as \]
where:
- \( a = g = 10 \, \text{m/s}^2 \) (acceleration due to gravity),
- \( s = 5 \, \text{m} \) (distance fallen).

Substituting the values:
\[ v^2 = 0 + 2 \times 10 \times 5 \]
\[ v^2 = 100 \]
\[ v = \sqrt{100} = 10 \, \text{m/s} \]

### Step 2: Determine the time taken to fall to the water surface
Using the equation:
\[ s = ut + \frac{1}{2}at^2 \]
Substituting the known values:
\[ 5 = 0 \cdot t + \frac{1}{2} \cdot 10 \cdot t^2 \]
This simplifies to:
\[ 5 = 5t^2 \]
\[ t^2 = 1 \]
\[ t = 1 \, \text{s} \]

### Step 3: Analyze the sinking motion from the water surface to the bottom
- The time taken to reach the bottom of the lake is given as 3 seconds.
- The time taken to fall to the water surface is 1 second, so the time taken to sink to the bottom is:
\[ t_{OP} = 3 \, \text{s} - 1 \, \text{s} = 2 \, \text{s} \]

### Step 4: Calculate the depth of the lake
- The ball sinks with a constant velocity (v = 10 m/s) from the water surface to the bottom.
- The distance (depth OP) can be calculated using:
\[ \text{Distance} = \text{Velocity} \times \text{Time} \]
Substituting the values:
\[ OP = 10 \, \text{m/s} \times 2 \, \text{s} = 20 \, \text{m} \]

### Final Answer
The depth of the lake is **20 meters**.

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