1. Numerical Approximation
For a spring-mass system the differential equation of the motion is:
If you know enough calculus (or know somebody who does) the solution is:
Where:

2. An Alternative to Doing All That Calculus
From the mass’ current position x calculate its acceleration: a = -(k / m) x
If the speed of the mass is v, calculate a new speed at a time Dt later: vnew = v + a Dt
If the position of the mass is x, calculate a new position at a time Dt later: xnew = x + vnew Dt
Go back to #1 and repeat
This is called Numerical Approximation (or Numerical Integration)

3. About Numerical Approximation
The solutions are only approximate
They can be made a close to correct as we wish by making the time step Dt small
Some systems, particularly chaotic ones can not be solved analytically
For such systems, numerical approximation is the only way that they can be solved

4. Numerical Approximation Module
We have prepared a working program that solves the spring mass system:
Using numerical approximation
Using the solution to the differential equation
You will “de-construct” the code to figure out how it works
The program is written in the Python language using the VPython visual library