40.3. From point $A$ along the circumference, two bodies start moving simultaneously in opposite directions: the first with a constant speed $v$, and the second with a constant linear acceleration $a$ and an initial speed of 0. After what time did the bodies meet for the first time, if they met again for the second time at point $A$?
40.3. Let's denote the length of the circumference by $l$. Suppose the time until the first meeting of the bodies is $t_{1}$, and the time until the second meeting is $t_{2}$. According to the problem, we have three equations:

$$
v t_{1}+\frac{a t_{1}^{2}}{2}=l, v t_{2}=l, \frac{a t_{2}^{2}}{2}=l
$$

By expressing $t_{2}$ from the second equation and substituting it into the third, we get: $l=2 v^{2}: a$. Now the first equation will take the form:

$$
\begin{aligned}
& v t_{1}+\frac{a t_{1}^{2}}{2}=\frac{2 v^{2}}{a}, \text { i.e. } a^{2} t_{1}^{2}+2 a v t_{1}-4 v^{2}=0, \\
& \\
\text { from which } t_{1}= & \frac{-a v+\sqrt{a^{2} v^{2}+4 a^{2} v^{2}}}{a^{2}}=\frac{v}{a}(\sqrt{5}-1) .
\end{aligned}
$$