Exercises

38.6. Exercises#

Exercise 38.1 (Independent and dependent)

Consider a free-falling body with mass \(m\). Neglecting a drag force it is straight-forward to use Newton’s equation to derive an expression for the distance \(d\) that the body has fallen during a time \(t\) when starting from rest at \(t=0\). Identify the dependent and the independent variable in this relation. What is the model parameter(s)?

Exercise 38.2 (Linear or non-linear)

Consider the relation between fall time \(t\) and velocity \(v\) for a free-falling body of mass \(m\) (starting from rest) that experiences a drag force that is modeled as \(bv\)

\[ v = v_T \left( 1 - e^{-\frac{b}{m}t}\right), \]

where \(v_T\) is the terminal velocity.

  • What are the model parameters?

  • Is this a linear or a non-linear model?

  • How would the relation look like if the drag force was neglected? Would that be a linear or a non-linear model?

Exercise 38.3 (Linear or non-linear; more examples)

Are these models linear or non-linear?

  1. \(\model{\pars}{\inputt} = \para_0 + (\para_1 \inputt)^2\)

  2. \(\model{\pars}{\inputt} = e^{\para_0 - \para_1\inputt/2}\)

  3. \(\model{\pars}{\inputt} = \para_0 e^{-\inputt/2}\)

  4. \(\model{\pars}{\inputt} = \para_0 e^{-\inputt/2} + \para_1 \sin(\inputt^2\pi)\)

  5. \(\model{\pars}{\inputt} = (\para_0 + \para_1 \inputt)^2\)

  6. \(\model{\pars}{\inputt} = (\para_0 + \para_1 \inputt)^2 + \para_2\inputt\)

Exercise 38.4 (Model discrepancy)

Consider again the relation between fall time \(t\) and velocity \(v\) for a free-falling body of mass \(m\) (starting from rest) that experiences a drag force that is modeled as \(bv\)

\[ v = v_T \left( 1 - e^{-\frac{b}{m}t}\right), \]

where \(v_T\) is the terminal velocity. Discuss possible model discrepancies. Are they expected to be large or small effects?