12 ODEs: Applications
18 Mar 2021After implementing a number of integrators for ODEs we are now in the position to solve some practical problems:
- projectile movement with air resistance
- realistic movement of a spinning baseball
Linear air resistance
For the problem that you should solve during the class, get the notebook1 12-ODE-lineardrag-students.ipynb and see notebook 12-ODE-lineardrag.ipynb for the full solution.2
Baseball physics
The major forces acting on a flying baseball are drag (friction due to air resistance) (see also 12-ODE-baseball.ipynb):
\[\mathbf{F}_2 = -b_2 v \mathbf{v}\]and the Magnus force (when the ball is spinning)
\[\mathbf{F}_M = \alpha \boldsymbol{\omega} \times \mathbf{v}\]where \(\boldsymbol{\omega}\) is the ball's angular velocity in rad/s (e.g., 200/s for a baseball).
For a sphere the proportionality constant \(\alpha\) can be written
\[\mathbf{F}_M = \frac{1}{2} C_L \rho A \frac{v}{\omega} \boldsymbol{\omega} \times \mathbf{v}\]where \(C_L\) is the lift coefficient, \(\rho\) the air density, \(A\) the ball's cross section. (Advantage of defining \(C_L\) this way: when spin and velocity are perpendicular, the Magnus force is simply \(F_M = \frac{1}{2} C_L \rho A v^2\).)
\(C_L\) is mainly a function of the spin parameter
\[S = \frac{r\omega}{v}\]with the radius \(r\) of the ball. In general we write
\[\mathbf{F}_M = \frac{1}{2} C_L \frac{\rho A r}{S} \boldsymbol{\omega} \times \mathbf{v}\]For a baseball, experimental data show approximately a power law dependence of \(C_L\) on \(S\)
\[C_L = 0.62 \times S^{0.7}\]The equations of motions are then
\begin{align} \frac{d\mathbf{r}}{dt} &= \mathbf{v}\\% \frac{d\mathbf{v}}{dt} &= -g \hat{\mathbf{e}}_y \mathbf{v} -\frac{b_2}{m} v \mathbf{v} + \alpha\ \boldsymbol{\omega} \times \mathbf{v} \end{align}
(quadratic drag \(-\frac{b_2}{m} v \mathbf{v}\) included.)
The baseball problem was live-coded from scratch during class and the notebook baseball_solution.ipynb2 is a cleaned-up and commented solution. The lesson showed how
- to define the physical problem (obtain trajectory of a baseball with spin and air resistance)
- to derive the underlying equations (Newton's equations of motions and the forces acting on the ball)
- to adapt an algorithm to solve the equations (use RK4 for
integrating the ODEs and express the problem in ODE standard form
so that one can use
ode.rk4()
) -
to visualize and discuss the results (plot trajectories and assess the influence of spin)
Resources
- Computational Modelling: Chapter 3
Footnotes
-
As usual,
git pull
the resources repository PHY494-resources to get a local copy of the notebook. Then copy the notebook and all other code into your work directory in order to complete the exercises. ↩ -
Notebook will be posted after class; in the meantime look at the student notebook. ↩ ↩2