Numerical Differentiation using Numerical Methods

Question

I am currently studying Numerical Differentiation in MATLAB using Numerical Methods in Engineering with Matlab by Jaan Kiusalaas, and I am stuck at exercise 13 from Problem Set 5.1 from Chapter 5 which goes like this:

The radar stations A and B, separated by the distance a = 500m, track the plane C by recording the angles α and β at one-second intervals. If three successive readings are

t (s) 9 10 11

α 54.80◦ 54.06◦ 53.34◦

β 65.59◦ 64.59◦ 63.62◦

calculate the speed v of the plane and the climb angle γ at t = 10 s. The coordinates of the plane can be shown to be

x = a * tan β / (tan β − tan α)

y = a * (tan α * tan β) / (tan β − tan α)

This is the drawing:

Can someone help me solve this by hand? No need for Matlab code.

Answer 1

To calculate the angle:

$\gamma=\tan^{-1}(\frac{y(11)-y(9)}{x(11)-x(9)})$

Where for instance $y(9)$ is the y value calculated at $t=9$ using the tan formula you have been given.

Then the speed at time $t=10$ would be given by:

$v(10)=(\sqrt{x(10)^2+y(10)^2})/10$

Answer 2

If $$\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \frac{a \tan\beta } { \tan \beta - \tan \alpha } \\ \frac{a \tan\alpha \tan\beta } { \tan \beta - \tan \alpha } \end{pmatrix} $$

then

$$\begin{pmatrix} \dot x \\ \dot y \end{pmatrix} = \begin{pmatrix} \frac{a (\dot\alpha \sin\beta \cos\beta - \dot \beta\sin \alpha \cos \alpha)}{\sin^2 (\alpha-\beta)} \\ \frac{a (\dot\alpha \sin^2\beta - \dot \beta\sin^2 \alpha )}{\sin^2 (\alpha-\beta)} \end{pmatrix} $$

So run your data through the cubic spline routine to get $\dot \alpha = \frac{\rm d}{{\rm d}t}\alpha$ and $\dot \beta=\frac{\rm d}{{\rm d}t}\beta$ at whatever time you need and use the above to get to the linear velocity components.

The above is derived by

$$\vec{r} = r(\alpha,\beta) \\ \dot{\vec{r}} = \vec{v} = \frac{\partial r}{\partial \alpha} \dot \alpha + \frac{\partial r}{\partial \beta} \dot \beta = v(\alpha,\beta,\dot \alpha,\dot \beta) $$