47. In congested traffic areas, to ensure traffic safety, the distance $d$ (meters) between motor vehicles and the speed $v$ (km/h) must follow the relationship $d \geqslant \frac{1}{2500} a v^{2}$ (where $a$ (meters) is the vehicle length, and $a$ is a constant), and it is also stipulated that $d \geqslant \frac{a}{2}$.
(1) When $d=\frac{a}{2}$, find the range of changes in the speed of motor vehicles.
(2) Let the hourly traffic flow of motor vehicles $Q=\frac{1000 v}{a+d}$, what speed should be specified to maximize the hourly traffic flow $Q$?
47. (1) From $\frac{a}{2} \geqslant \frac{1}{2500} a v^{2}$, we get $025 \sqrt{2}$, $Q=-\frac{1000}{a\left(\frac{1}{v}+\frac{v}{2500}\right)} \leqslant \frac{25000}{a}$,
$\therefore$ When $v=50$, $Q$ is maximized at $\frac{25000}{a}$.