How Deep Is the Well?

How Deep Is the Well? Exploring Physics with Code

Have you ever dropped a rock into a well and wondered how deep it is? The sound of the splash echoes back, but how can we calculate the depth without physically measuring it? Let’s dive into the fascinating world of physics and programming to solve this age-old question using Function Calculator. 

The Science Behind It

When you drop an object into a well, two key events occur:

1. The Fall: The object falls under the influence of gravity, accelerating as it goes.

2. The Sound: After hitting the water, the sound of the splash travels back up to you.

By combining these two events, we can calculate the depth of the well using basic physics. The total time t (the time it takes for you to hear the splash) is a combination of:

• The time it takes for the object to fall (fallTime).

• The time it takes for the sound to travel back up (soundTime).

Using these principles, we can derive a quadratic equation to calculate the depth.

The Formula

The physics formula boils down to solving this equation:

t=fallTime+soundTime

Where:

• fallTime=sqrt(2*d/g) (time for the object to fall)

• soundTime=v/d (time for sound to travel back)

• Where sqrt is square root and * is multiplication operator and / division operator.

Here:

• g=9.8m/s^2 (acceleration due to gravity)

• v=343m/s (speed of sound in air)

• d is the depth of the well.

Substituting these into one equation gives us a quadratic formula:

(1/2)*g*x^2+v*x−t=0

Where x=fallTime. Solving this equation gives us x, which we use to find d.

Coding It

Here’s how we can turn this physics problem into code:

If you do not have Function Calculator the first install it.

Copy below code and go to the Factory page  in the calculator and paste it into the editor.

And click run button to execute the code and click save button to save below two functions.

function solveQuadratic(a, b, c, x12) {
  var D = b * b - 4 * a * c;
  if (D < 0) {return -1;}
  var sqrtD = sqrt(D);
  if(x12 == 1){
    return (-b + sqrtD) / (2 * a);}
  else{return (-b - sqrtD) / (2 * a);}
}

function depth(t) {
  var g = 9.8;
  var sound = 343;
  
  var fallTime = solveQuadratic(0.5 * g / sound, 1, -t, 1);
  
  if (fallTime== -1) {return -1;}
  
  var depth = 0.5 * g * pow(fallTime, 2);
  
  return depth;
  }
  

  
    
  

How It Works

1. Input: You provide the total time t, which is how long it took for you to hear the splash after dropping an object.

2. Calculation:

   • The function solves the quadratic equation to find fallTime.

   • Using fallTime, it calculates the depth of the well.

3. Output: The function returns the depth in meters.

Example in Action

Let’s say you drop a rock into a well and hear the splash after 2 seconds. 

Go to the calculator page an to select 'depth(t)' in the list.

You can calculate like below picture.

The well is approximately 19 meters deep. 

 If 5 seconds then the depth of well is as below. 107 meteres deep! That's quite a drop!

With Function Calculator you can solve many interesting mathematic problems.