Engineering at Alberta Courses

Numerical Differentiation: Problems

Compare the basic formulas and the high-accuracy formulas to calculate the first and second derivatives of $e^{x}$ at $x=2$ with $h=0.1$ . Clearly indicate the order of the error term in the approximation used. Calculate the relative error $E_r$ in each case.
A plane is being tracked by radar, and data are taken every two seconds in polar coordinates $\theta$ and $r$ .

Time $t$ (sec.) 200 202 204 206 208 210

$\theta$ (rad) 0.75 0.72 0.70 0.68 0.67 0.66

$r$ (m.) 5120 5370 5560 5800 6030 6240

The vector expression in polar coordinates for the velocity $v$ and the acceleration $a$ are given by:

$\begin{split} v&=\frac{\mathrm{d}r}{\mathrm {d}t}e_r+r\frac{\mathrm{d}\theta}{\mathrm {d}t}e_{\theta} \\ a&=\left(\frac{\mathrm{d}^2r}{\mathrm {d}t^2}-r\left(\frac{\mathrm{d}\theta}{\mathrm {d}t}\right)^2\right)e_r+\left(r\frac{\mathrm{d}^2\theta}{\mathrm {d}t^2}+2\frac{\mathrm{d}r}{\mathrm {d}t}\frac{\mathrm{d}\theta}{\mathrm {d}t}\right)e_{\theta} \end{split}$

Use the centred finite difference basic formulas to find the velocity and acceleration vectors at $t = 206$ sec. as a function of the unit vectors $e_r$ and $e_{\theta}$ . Then, using the norm of the velocity and acceleration vectors describe the plane speed and the magnitude of its acceleration.
Fick’s first diffusion law states that:
$\mbox{Mass Flux}=-D\frac{\mathrm{d}C}{\mathrm{d}x}$

where $\mbox{Mass Flux}$ is the mass transported per unit area and per unit time with units $g/m^2/s$ , $D$ is a constant called the diffusion coefficient given in $m^2/sec.$ , $C$ is the concentration given in $g/m^3$ and $x$ is the distance in $m$ . The law states that the mass transported per unit area and per unit time is directly proportional to the gradient of the concentration (with a negative constant of proportionality). Stated differently, it means that the molecules tend to move towards the area of less concentration. If the concentration of a pollutant in the pore waters of sediments underlying the lake is constant throughout the year and is measured to be:

$x (m)$ 0 1 3

$C$ , $10^{-6} (g/m^3)$ 0.06 0.32 0.6

where $x=0$ is the interface between the lake and the sediment. By fitting a parabola to the data, estimate $\frac{\mathrm{d}C}{\mathrm{d}x}$ at $x=0$ . If the area of interface between the lake and the sediment is $3.6\times 10^6 m^2$ , and if $D=1.52\times 10^{-6}m^2/sec$ , how much pollutant in $g$ would be transported into the lake over a period of one year?
The positions in m. of the left and right feet of two squash players (Ramy Ashour and Cameron Pilley) during an 8 second segment of their match in the North American Open in February 2013 are given in the excel sheet provided here. Using a centred finite difference scheme, compare the average and top speed and acceleration between the two players during that segment of the match. Similarly, using a parabolic spline interpolation function, compare the average and top speed and acceleration between the two players. Finally, compare with the average speed of walking for humans.