A stone is dropped into a well and is heard to strike the water after 4 second.Find the depth of..

Introduction

In this tutorial, we will determine the depth of a well using the time it takes for a stone dropped into the well to be heard striking the water. The total time for the stone to fall and the sound to travel back up is given as 4 seconds, and we will use the velocity of sound, which is 350 m/s, to help us calculate the depth.

Step 1: Understand the Problem

To find the depth of the well, we need to break the problem into two parts:

• The time it takes for the stone to fall to the water (t1).
• The time it takes for the sound to travel back up to the top of the well (t2).

Since the total time is 4 seconds, we can express this relationship as:

• Total time = t1 + t2 = 4 seconds

Step 2: Define the Variables

Let’s set up the variables:

• Let h be the depth of the well in meters.
• t1 = time taken for the stone to fall to the water.
• t2 = time taken for the sound to travel back up to the top.

Using the speed of sound:

• t2 can be calculated using the formula: [ t2 = \frac{h}{v} ] where v is the velocity of sound (350 m/s).

Step 3: Establish the Equation for Falling Stone

For the stone falling, we can use the formula for distance: [ h = \frac{1}{2} g t1^2 ] where g is the acceleration due to gravity (approximately 9.81 m/s²).

Step 4: Set Up the Total Time Equation

Now we substitute t2 into the total time equation: [ t1 + \frac{h}{350} = 4 ]

Step 5: Substitute h into the Total Time Equation

From the equation of the falling stone, we can express h in terms of t1: [ h = \frac{1}{2} g t1^2 ]

Now substitute h back into the total time equation: [ t1 + \frac{\frac{1}{2} g t1^2}{350} = 4 ]

Step 6: Solve for t1

Substituting g = 9.81 m/s², we get: [ t1 + \frac{4.905 t1^2}{350} = 4 ] To simplify this, multiply through by 350 to eliminate the fraction: [ 350 t1 + 4.905 t1^2 = 1400 ] Rearranging gives: [ 4.905 t1^2 + 350 t1 - 1400 = 0 ]

Step 7: Use the Quadratic Formula

Now we can solve this quadratic equation using the quadratic formula: [ t1 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ] Where:

• a = 4.905
• b = 350
• c = -1400

Substituting these values into the formula will yield the time t1.

Step 8: Calculate Depth of the Well

Once you find t1, substitute it back into the equation for h: [ h = \frac{1}{2} g t1^2 ] This will give you the depth of the well in meters.

Conclusion

By following these steps, you can calculate the depth of a well based on the time taken for a stone to hit the water and for the sound to travel back. The key takeaway is to break the problem into manageable parts and apply relevant physics formulas. For further practice, consider varying the velocity of sound or the total time to see how it affects your results.