The problem can be diagrammatically represented as follows:

Since its a parabola opening upwards,its equation is $x^2 = 4ay$.

The parabola passes through the point $ (\large\frac{5}{2}$$,10)$

Therefore, $\large(\frac{5}{2})^2 $$=4\times a \times 10 \rightarrow a = \large\frac{5}{32}$

Therefore, the equation of the parabola is $x^2 = 4\times\large\frac{5}{32}$$\times y = \large\frac{5y}{8}$

We have to solve for $y=2m \rightarrow x^2 = \large\frac{5 \times 2}{8}$$ \rightarrow x = \large\sqrt {\frac{5}{4}}$

$\qquad AB = 2x = 2 \large\sqrt {\frac{5}{4}}$$ \approx 2.23m$