When a ball is dropped into a lake from a height 4.9 m above the water level , it hits the water with a velocity v and then sinks to the bottom with the constant velocity v , It reaches the bottom of the lake 4.0 s after it is dropped . The approximately depth of the lake is :

Options

A. `19.6` m
B. `29.4`m
C. `39.2` m
D. `73.5` m

Explanation

To solve this problem, we need to determine the depth of the lake given the conditions provided. Let's break it down step by step:

### Step 1: Calculate the time taken to hit the water
First, we need to find the time it takes for the ball to fall from the height of 4.9 meters above the water level to the water surface.

Using the equation of motion:
\[ s = ut + \frac{1}{2}at^2 \]

Given:
- Initial velocity \( u = 0 \) (since the ball is dropped)
- Acceleration \( a = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)
- Distance \( s = 4.9 \, \text{m} \)

Substitute the values into the equation:
\[ 4.9 = 0 \cdot t + \frac{1}{2} \cdot 9.8 \cdot t^2 \]
\[ 4.9 = 4.9t^2 \]
\[ t^2 = 1 \]
\[ t = 1 \, \text{second} \]

So, it takes 1 second for the ball to hit the water.

### Step 2: Calculate the velocity with which the ball hits the water
Using the first equation of motion:
\[ v = u + at \]

Given:
- Initial velocity \( u = 0 \)
- Acceleration \( a = 9.8 \, \text{m/s}^2 \)
- Time \( t = 1 \, \text{second} \)

Substitute the values:
\[ v = 0 + 9.8 \cdot 1 \]
\[ v = 9.8 \, \text{m/s} \]

The ball hits the water with a velocity of \( 9.8 \, \text{m/s} \).

### Step 3: Calculate the time taken to reach the bottom of the lake
The total time taken to reach the bottom of the lake is given as 4 seconds. We already know it takes 1 second to reach the water surface. Therefore, the time taken to sink to the bottom of the lake is:
\[ \text{Time in water} = 4 \, \text{seconds} - 1 \, \text{second} = 3 \, \text{seconds} \]

### Step 4: Calculate the depth of the lake
Since the ball sinks with a constant velocity \( v = 9.8 \, \text{m/s} \), we can use the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]

Given:
- Speed \( v = 9.8 \, \text{m/s} \)
- Time \( t = 3 \, \text{seconds} \)

Substitute the values:
\[ \text{Depth} = 9.8 \times 3 \]
\[ \text{Depth} = 29.4 \, \text{meters} \]

Therefore, the depth of the lake is approximately \( 29.4 \, \text{meters} \).

### Final Answer:
The depth of the lake is \( \boxed{29.4 \, \text{meters}} \).