Another way to represent the system is in input-output form. In this representation, the inputs and their derivatives are on one side of the equation and the outputs and their derivatives are on the other side of the equation. In input-output form the system will look like

We can divide both sides by M to get a more standard equation

One way to represent the system is in state-variable notation. In this notation, we represent the energy storing elements of the system with state variables. We chose as the state variables the elongation x of the spring, which is related to its potential energy, and the velocity v of the mass, which is related to its kinetic energy. From inspection of the system and knowledge that the derivative of the displacement of a mass is its velocity, we get the first state-variable equation to be x’=v. The second state-variable equation is found by solving the system equation for v’ to get

Represented in matrix form these equations are

Since the goal is to determine how long to make the bungee cord so that the jumper doesn’t make contact with the ground, one output of interest is the elongation of the bungee cord x. Also of interest is the tensile strength of the bungee cord. We do not want a heavy jumper to exceed the rated tensile strength of the cord. For if he does, the cord may break and spell disaster for the jumper and the company. Represented in matrix form these equations are