One, (20 points) (1) Table 1 shows the data measured in a projectile experiment of an object.

Assuming the trajectory equation of the object is $y=a x^{2}$, find the value of the real number $a$ so that the sum of the squares of the distances from each measurement point $P_{i}\left(x_{i}, y_{i}\right)(i=1,2,3, 4)$ to the corresponding point $A_{i}\left(x_{i}, a x_{i}^{2}\right)$ on the trajectory curve $\sum_{i=1}^{4} P_{i} A_{i}^{2}$ is minimized.
(2) Under the requirement of (1), for the data in Table 2, find the equation of the trajectory curve $y=a x^{2}$ (the coefficient $a$ should be accurate to $0.01$).
(1) From the problem, we know
$$
\begin{array}{l}
\sum_{i=1}^{4} P_{i} A_{i}^{2}=\sum_{i=1}^{4}\left(a x_{i}^{2}-y_{i}\right)^{2} \\
=a^{2} \sum_{i=1}^{4} x_{i}^{4}-2 a \sum_{i=1}^{4} x_{i}^{2} y_{i}+\sum_{i=1}^{4} y_{i}^{2} \\
=\sum_{i=1}^{4} x_{i}^{4}\left[a^{2}-\frac{2 \sum_{i=1}^{4} x_{i}^{2} y_{i}}{\sum_{i=1}^{4} x_{i}^{4}} a+\left(\frac{\sum_{i=1}^{4} x_{i}^{2} y_{i}}{\sum_{i=1}^{4} x_{i}^{4}}\right)^{2}+\right. \\
\sum_{i=1}^{4} y_{i}^{2}-\left(\frac{\sum_{i=1}^{4} x_{i}^{2} y_{i}}{\sum_{i=1}^{4} x_{i}^{4}}\right)^{2} \\
=\sum_{i=1}^{4} x_{i}^{4}\left(a-\frac{\sum_{i=1}^{4} x_{i}^{2} y_{i}}{\sum_{i=1}^{4} x_{i}^{4}}\right)^{2}+\sum_{i=1}^{4} y_{i}^{2}-\left(\frac{\sum_{i=1}^{4} x_{i}^{2} y_{i}}{\sum_{i=1}^{4} x_{i}^{4}}\right)^{2} \\
\text { Therefore, when } a=\frac{\sum_{i=1}^{4} x_{i}^{2} y_{i}}{\sum_{i=1}^{4} x_{i}^{4}}, \sum_{i=1}^{4} P_{i} A_{i}^{2} \text { reaches its minimum value } \\
\sum_{i=1}^{4} y_{i}^{2}-\left(\frac{\sum_{i=1}^{4} x_{i}^{2} y_{i}}{\sum_{i=1}^{4}} x_{i}^{4}\right.
\end{array}
$$

Therefore, when $a=\frac{\sum_{i=1}^{4} x_{i}^{2} y_{i}}{\sum_{i=1}^{4} x_{i}^{4}}$, $\sum_{i=1}^{4} P_{i} A_{i}^{2}$ reaches its minimum value.
(2) From the data in Table 2, we get
$$
\begin{array}{l}
\sum_{i=1}^{4} x_{i}^{4}=1^{4}+2^{4}+3^{4}+4^{4}=354, \\
\sum_{i=1}^{4} x_{i}^{2} y_{i}=1^{2} \times(-2.6)+2^{2} \times(-9.9)+ \\
\quad 3^{2} \times(-22.6)+4^{2} \times(-39.8) \\
=-882.4 .
\end{array}
$$

Thus, from (1), we have $a=\frac{-882.4}{354} \approx-2.49$.
Therefore, the desired orbital curve is $y=-2.49 x^{2}$.