A lead ball is dropped into a lake from a diving board 5.70 m above the water. After entering the water, it sinks to the bottom with a constant velocity equal to the velocity with which it hit the water. The ball reaches the bottom 4.40 s after it is released. How deep is the lake?

To solve this problem, we can break it into two parts: 

1. **Find the velocity of the ball when it hits the water** (free fall).
2. **Calculate the depth of the lake** using the constant velocity with which the ball sinks.

### Step 1: Find the velocity at which the ball hits the water

The ball is dropped from a height of 5.70 m, so it is in free fall. We can use the following kinematic equation for the velocity \( v \) of the ball just before hitting the water:

\[
v = \sqrt{2gh}
\]

where:
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity),
- \( h = 5.70 \, \text{m} \) (height from which the ball is dropped).

Now we calculate the velocity \( v \).

\[
v = \sqrt{2 \times 9.8 \, \text{m/s}^2 \times 5.70 \, \text{m}} 
\]

### Step 2: Calculate the depth of the lake

The ball hits the water with velocity \( v \) and sinks to the bottom at the same constant velocity. The total time taken for the entire motion is 4.40 seconds, and the time taken for the free fall portion can be found by using the formula:

\[
t_{\text{fall}} = \sqrt{\frac{2h}{g}}
\]

Subtracting \( t_{\text{fall}} \) from the total time gives the time taken to sink \( t_{\text{sink}} \):

\[
t_{\text{sink}} = 4.40 \, \text{s} - t_{\text{fall}}
\]

Using this time, we can calculate the depth of the lake, since \( d = v \cdot t_{\text{sink}} \). Let's solve this step-by-step.### Results:

1. The velocity of the ball when it hits the water is approximately \( 10.57 \, \text{m/s} \).
2. The time taken to fall into the water is \( 1.08 \, \text{seconds} \).
3. The time taken for the ball to sink to the bottom of the lake is \( 3.32 \, \text{seconds} \).
4. The depth of the lake is approximately \( 35.11 \, \text{meters} \).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How would the depth of the lake change if the ball's velocity decreased when it entered the water?
2. What if the ball had been thrown downward instead of dropped from rest?
3. How long would the ball take to reach the bottom if it was released from a greater height?
4. What effect would different gravitational forces (e.g., on other planets) have on the time to reach the bottom?
5. How would the result change if the ball experienced deceleration in the water?

**Tip:** Always break down problems with multiple stages (free fall, constant velocity) by analyzing each phase separately!

A lead ball is dropped into a lake from a diving board 5.70 m above the water. After entering the water, it sinks to the bottom with a constant velocity equal to the velocity with which it hit the water. The ball reaches the bottom 4.40 s after it is released. How deep is the lake?

Solve how deep a lake is by analyzing a lead ball's motion dropped from a height of 5.70 m. This physics problem covers key concepts like free fall and constant velocity motion, using kinematic equations to find the velocity and time involved. Suitable for students in Grades 10-12.

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