Math Problem Statement

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?

Solution

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 vv of the ball just before hitting the water:

v=2ghv = \sqrt{2gh}

where:

  • g=9.8m/s2g = 9.8 \, \text{m/s}^2 (acceleration due to gravity),
  • h=5.70mh = 5.70 \, \text{m} (height from which the ball is dropped).

Now we calculate the velocity vv.

v=2×9.8m/s2×5.70mv = \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 vv 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:

tfall=2hgt_{\text{fall}} = \sqrt{\frac{2h}{g}}

Subtracting tfallt_{\text{fall}} from the total time gives the time taken to sink tsinkt_{\text{sink}}:

tsink=4.40stfallt_{\text{sink}} = 4.40 \, \text{s} - t_{\text{fall}}

Using this time, we can calculate the depth of the lake, since d=vtsinkd = 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 10.57m/s10.57 \, \text{m/s}.
  2. The time taken to fall into the water is 1.08seconds1.08 \, \text{seconds}.
  3. The time taken for the ball to sink to the bottom of the lake is 3.32seconds3.32 \, \text{seconds}.
  4. The depth of the lake is approximately 35.11meters35.11 \, \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!

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Math Problem Analysis

Mathematical Concepts

Kinematics
Free Fall
Constant Velocity

Formulas

v = √(2gh)
t_fall = √(2h/g)
d = v * t_sink

Theorems

Kinematic equations for uniformly accelerated motion
Constant velocity motion

Suitable Grade Level

Grades 10-12