5. (10 points) A small railway wagon with a jet engine is standing on the tracks. The tracks are laid in the form of a circle with a radius of $R=4$ m. 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 $a=2 \mathrm{M} / \mathrm{c}^{2}$?
Answer: $\approx 2 \mathrm{m} / \mathrm{s}$.

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

In addition, the cart has a centripetal acceleration:

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

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

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

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

As a result:

$$
v_{\max } \approx 2.8 \mathrm{~m} / \mathrm{s}
$$

Answer: $\approx 2.8 \mathrm{m} / \mathrm{s}$.