69. A reconnaissance aircraft is flying in a circle centered at point $A$. The radius of the circle is $10 \mathrm{km}$, and the speed of the aircraft is 1000 km/h. At some moment, a missile is launched from point $A$ with the same speed as the aircraft, and it is controlled such that it always remains on the line connecting the aircraft to point $A$. How long after the launch will the missile catch up to the aircraft?
69. An s w e r: in $\frac{\pi}{200}$ h. It is necessary to notice that the trajectory of the imaginary "rocket" in the conditions of the problem is a circle of half the radius (Fig. 46): the degree measure of the arc $A R$ is twice that of $\angle Q A P$ (the angle between the tangent and the chord), i.e., twice the degree measure of the arc $\widetilde{Q P}$ of twice the radius; therefore, the lengths of these arcs (for each position of $R$) are equal. Before the meeting, the plane will travel a quarter of the circumference, the rocket - half.

$\nabla$ It would be necessary, however, to prove the uniqueness of the rocket's trajectory satisfying the condition (this was not required of the olympiad participants: at that time, derivatives were not studied in school). It follows relatively easily from the general theorem on the uniqueness of a function with a given value at $t=0$ and a given derivative; if at time $t$ the rocket has traveled a distance $A R=A Q f(t)$ from point $A$ (where $f(0)=0$), then its velocity in the direction perpendicular to the radius is $v f(t)$, and along the radius $-v \sqrt{1-f^{2}(t)}=f(t)$, so $\frac{f^{\prime}(t)}{\sqrt{1-f^{2}(t)}}=v$, i.e., $(\arcsin f(t))^{\prime}=v$, from which $f(t)=\sin v t$.