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:

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)

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

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