This deceptively simple question turns into a classic engineering puzzle involving gravity, sound propagation and iterative solving. In this article, we walk through how engineers can model and solve interdependent equations using Mathcad — turning a “back-of-the-envelope” physics riddle into a clear and auditable engineering calculation. Along the way, we’ll see why this type of problem shows up far more often in real engineering work than you might expect.
The Scenario
A stone is dropped into a vertical shaft.
You measure the total time until you hear it hit the bottom.
The depth of a mine shaft can be calculated by timing how long it takes a rock to fall (neglecting air resistance and the time it takes for the sound to get from the bottom of the shaft up to our ears) by using the equation: d = ½a*t² where a is the acceleration due to gravity and t is the fall time.
We know that the acceleration of gravity is 9.807 m/s². Once we know the time, we can calculate the depth of the shaft.
Creating User-Defined Functions with Mathcad
We will create a function in Mathcad called “Depth” with a variable of time (t). Simply define the variable and then list the inputs to the function in parenthesis and then tell Mathcad the equation using the input variables.
You can type the equation and put the value of t in parenthesis. Then, attach the units of seconds. Now you can test several variables of time and assign a variable to the results of the function.
This illustrates the ease of user-defined functions in Mathcad.
If we now include the time it takes the sound to travel from the bottom of the shaft to our ears, the calculation requires some iteration to find the depth of the shaft.
We will write an equation for how long it takes the sound to get to your ears as a function of depth.
The first part of the equation will calculate the time it takes for the rock to hit the bottom of the shaft and the second part of the equation will calculate how long it takes for the sound to get from the bottom of the shaft to reach your ears.
Using the Solve function in Mathcad
To solve the problem, create a function that will provide a solution using the solve keyword from the symbolic engine.
The time it takes for sound to get to the surface is the depth of the shaft divided by the velocity of sound.
Creating a Function in Mathcad
To solve the problem, create a function that will provide a solution using the solve keyword from the symbolic engine.
Test the function for a measured time of 8 seconds.
The larger value comes from the negative value of time and can be ignored. Therefore, modify the function to select the minimum value of depth by using the min function.
Now that you have created a function to calculate the depth of the shaft, you can simply enter the measured time and get a direct solution of shaft depth.
Why Engineers Use Mathcad for Problems Like This
This example mirrors many real engineering tasks:
With Mathcad, engineers can:
If your work involves complex calculations, iterative solving or technical documentation, Mathcad is built for exactly that.
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Calculations are at the heart of engineering. PTC Mathcad is mathematical software that lets you easily perform, analyse, document and share your calculations.