42.7. Solutions#
Solution to Exercise 42.1 (Independent and dependent)
The relation is \(d = g t^2/2\). Here we are describing how the distance traveled depends on the time of the free fall. Therefore \(d\) is the dependent variable and \(t\) is the independent one. There is a single model parameter \(g\) that we could infer from observational data.
Solution to Exercise 42.2 (Linear or non-linear)
The model parameters are \(v_T\) and \(b/m\). Alternatively, since \(v_T = mg/b\), we could conisder \(g\) and \(b/m\) as the model parameters. It would not be correct to claim that we have three model parameters since \(b\) and \(m\) only appear in a ratio.
This is a non-linear model since \(b/m\) appears in an exponential.
The corresponding relation without drag force is \(v = gt\). That is a linear model.
Solution to Exercise 42.3 (Linear or non-linear; more examples)
Linear (we can consider \(\para_1^2\) as a parameter).
Non-linear.
Linear (there is no parameter-dependence in the exponential function).
Linear.
Non-linear. It would be tempting to consider \(\para_0^2\), \(\para_1^2\), and \(2\para_0 \para_1\) as three independent parameters in which case it would be a linear model. But these are not independent and we would need to keep the quadratic parameter dependence.
Linear if we consider \(\para_0^2\), \(\para_1^2\), and \(2\para_0 \para_1 + \para_2\) as parameters.
Solution to Exercise 42.4 (Model discrepancy)
First, we are assuming that the gravitational force is constant for the duration of the fall. This approximation corresponds to setting \(GM/(R+h)^2 \approx GM / R^2 = g\) where \(G\) is the gravitational constant, \(M\)(\(R\)) is the earth mass(radius), and we neglect the small and varying height \(h\). The error that is made here will be of order \((h/R)^2\) which is really small.
More importantly, the linear drag force is a simplification. We could add a term that is quadratic in the velocity. The error made by not including this term will grow with velocity.
Finally, Newton’s equations of motion has turned out to be an approximation of the general theory of relativity. Again, the modeling error will be negligible for “normal” masses and velocities.