An object was lanched from a building and had its height registered by the following table:
Height 192 180 150 115 72
Time --- 1 - 2 --- 3 -- 4--- 5
Use the least squares method to estimate the height, $g$ and the vertical velocity at $t=0$
I think I must fit a curve into this dataset. I'm studying orthogonal families of polynomials and projection onto subspaces, so I guess I should use these concepts.
The first question is: which function I should use to approximate to the dataset and how many free parameters I should use?
Assuming that the dynamical model is
$$ \ddot y = -g $$
we have as general solution
$$ y(t) = -\frac 12 g t^2+c_1 t + c_2 $$
so defining now the measurement error as
$$ e_k^2 = \left(y(t_k) + y_k\right)^2 $$
and calling the total error as
$$ E_2 = \sum_k e_k^2 = f(c_1,c_2,g) $$
the minimum total error is attained at
$$ \frac{\partial E_2}{\partial c_1} = \frac{\partial E_2}{\partial c_2} = \frac{\partial E_2}{\partial g} = 0 $$
now solving the resulting linear system we get
$$ c_1 = -\frac{25}{14},\ \ c_2 = \frac{999}{5},\ \ g = \frac{67}{7} $$
now at $t = 0$ we have $y_0 = c_2 = \frac{999}{5}$ and $\dot y_0 = v_0 = c_1 =-\frac{25}{14}$
Attached a plot showing in blue the adjusted function $y(t)$ and in red the measuring points.