91Ó°ÊÓ

A plane is being tracked by radar, and data is taken every second in polar coordinates \(\theta\) and \(r\) At 206 seconds, use the centered finite difference (second-order correct) to find the vector expressions for velocity \(\vec{v},\) and acceleration \(\bar{a} .\) The velocity and acceleration given in polar coordinates are: \\[\bar{v}=\dot{r} \bar{e}_{r}+r \dot{\theta} \bar{e}_{\theta} \quad \text { and } \quad \bar{a}=\left(\ddot{r}-r \dot{\theta}^{2}\right) \bar{e}_{r}+(r \ddot{\theta}+2 \dot{r} \dot{\theta}) \bar{e}_{\theta}\\]

Use regression to estimate the acceleration at each time for the following data twith second-, third-, and fourth-order polynomials. Plot the results $$\begin{array}{c|cccccccccc} ! & 1 & 2 & 3.25 & 4.5 & 0 & 7 & 8 & 8.5 & 9.3 & 10 \\ \hline v & 10 & 12 & 11 & 14 & 17 & 16 & 12 & 14 & 14 & 10 \end{array}$$

You have to measure the flow rate of water through a small pipe. In order to do it, you place a bucket at the pipe's outlet and measure the volume in the bucket as a function of time as tabulated below. Estimate the flow rate at \(t=7 \mathrm{s}\) $$\begin{array}{l|cccc} \text { Time, 5 } & 0 & 1 & 5 & 8 \\ \hline \text { Volume, } \mathrm{cm}^{3} & 0 & 1 & 8 & 16.4 \end{array}$$

Chemical reactions often follow the model: \frac{d c}{d t}=-k c^{n} where \(c=\) concentration, \(t=\) time. \(k=\) reaction rate, and \(n=\) reaction order. Given values of \(c\) and \(d c / d t, k\) and \(n\) can be evaluated by a linear regression of the logarithm of this equation: $$\log \left(-\frac{d c}{d t}\right)=\log k+n \log c$$ Use this approach along with the following data to estimate \(k\) and \(n\) : $$\begin{array}{c|cccccc} I & 10 & 20 & 30 & 40 & 50 & 60 \\ \hline c & 3.52 & 2.48 & 1.75 & 1.23 & 0.87 & 0.61 \end{array}$$