5. (10 points) A small railway wagon with a jet engine is standing on the tracks. The tracks are laid out in the form of a circle with radius $R$. The wagon starts from rest, with the jet force having a constant value. What is the maximum speed the wagon will reach after one full circle, if its acceleration over this period should not exceed the value $a$?
Answer: $v_{\max }=\sqrt[4]{\frac{16 a^{2} R^{2} \pi^{2}}{\left(1+16 \pi^{2}\right)}}$

Solution. The acceleration of the trolley: $a_{1}=\frac{v^{2}}{2 s}=\frac{v^{2}}{4 \pi R}$.

In addition, the trolley has a centripetal acceleration:

$a_{2}=\frac{v^{2}}{R}$.

The total acceleration of the trolley: $a^{2}=a_{1}^{2}+a_{2}^{2}=\left(\frac{v^{2}}{4 \pi R}\right)^{2}+\left(\frac{v^{2}}{R}\right)^{2}$.

We obtain that the final speed of the trolley cannot be greater than:

$$
v_{\max }=\sqrt[4]{\frac{16 a^{2} R^{2} \pi^{2}}{\left(1+16 \pi^{2}\right)}}
$$