4. A little rascal ran a race with a mechanical horse on a carousel. After a seconds, she discovered that the horse, having made a full circle, had caught up with her. Instantly turning around, the little rascal ran towards the horse at the same speed and met it after \(\frac{a}{2}\) seconds. Determine how long it takes for the carousel to make a full revolution, assuming all movements are uniform.
# Solution.

Let $v$ be the angular velocity of the hedgehog, $U$ be the angular velocity of the carousel, and $x$ be the period of the carousel. Note that $\frac{a}{2}<x<a$. The distance the hedgehog ran in $a$ seconds is the same distance the carousel traveled in $a-x$ seconds. Therefore, we have

$$
v a=U(a-x)
$$

After turning around, the distance the hedgehog ran in $\frac{a}{2}$ seconds is the same distance the carousel would have traveled in $x-\frac{a}{2}$ seconds, so

$$
\frac{v a}{2}=U\left(x-\frac{a}{2}\right)
$$

From this, we get the equation

$$
\frac{a-x}{a}=\frac{2 x-a}{a}
$$

Solving for $x$, we find $x=\frac{2}{3} a$.

Answer. The carousel completes one full revolution in $\frac{2}{3} a$ seconds.