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3.4 Working with models

Example: Free fall

Course element	Work	Learning activity
Elaboration Construction of model lines and computation	Elaboration of the model in mathematical terms or in a graphical model representation Educated guess of parameter values in the model on the basis of data Computation of the model with an adequate time step	Class discussion Group work

The computation of the falling motion in subsequent time steps, as elaborated in Table 2, can be systematised in an iterative calculation scheme.¹

Model lines of code for free fall
1. Change of velocity	$Δ v = - g \cdot Δ t$
2. New velocity	$v : = v + Δ v$
3. Change of position	$Δ y = v \cdot Δ t$
4. New position	$y : = y + Δ y$
5. New time	$t : = t + Δ t$

TABLE 3. Iterative calculation scheme of free fall

In the first two model lines, a new velocity is calculated from the acceleration; hereafter a new position is calculated from this velocity. The last model line adds one time step to the time for a new calculation. These calculation steps can be repeated until a chosen final situation is reached, for example y = 0. The change of a state of this dynamic model can thus be fully expressed in terms of the acceleration and the initial states of position and velocity (or end states when one calculates back in time).

With the model lines in Table 3, the falling motion of an object subject to gravity only can be calculated on a computer. A specific modelling environment is usually used for this, such as Coach² or Modellus.³ For a simple phenomenon such as free fall, the calculations can also be performed in a spreadsheet program such as Excel.⁴

The sequence of the steps in the calculation scheme is important.⁵ First the new velocity is calculated and then the new location. This has to do with the fact that the velocity is not constant over a time interval Δt; with a uniformly accelerated motion, the velocity increases linearly in time. In fact, the average velocity should be calculated:
$v_{avg} = \frac{1}{2} (v_{-} + v_{+}) = v_{-} + \frac{1}{2} a \cdot Δ t = v_{+} - \frac{1}{2} a \cdot Δ t$
Here v_- and v₊ are the initial velocity and final velocity, respectively, for a given time step.

In the calculation scheme in Table 3 the following approximation is chosen: $v_{avg} ≅ v_{+}$ . In numerical mathematics this is known as the Euler-Cromer algorithm. This algorithm differs from the commonly used numerical method called the Euler method, which is based on the approximation $v_{avg} ≅ v_{-}$ . Thus, the Euler method computes with the initial velocity for a given time interval and the Euler-Cromer method with the final velocity. Both methods have an approximation error of order Δt over a finite interval, but the Euler-Cromer method is more stable.⁶

An even better approach for the numerical iteration of equations of motion is the so-called leapfrog method, as described in the Feynman Lectures⁷, which basically means that the formula for the above average velocity is numerically implemented.⁸