Let's set the time when the car passes the perpendicular point drawn from the locator to the road as 0! We measure the angular displacement of the locator from this perpendicular!

![](https://cdn.mathpix.com/cropped/2024_05_02_271f369abba9e95c27d9g-1.jpg?height=355&width=326&top_left_y=278&top_left_x=886)

Figure 1

Based on Figure 1:

$$
\operatorname{tg} \alpha=\frac{v \cdot t}{h} ; \quad t=\frac{h}{v} \operatorname{tg} \alpha
$$

Thus,

$$
\alpha=\operatorname{arctg}(v \cdot t / h)
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_271f369abba9e95c27d9g-1.jpg?height=371&width=539&top_left_y=1105&top_left_x=782)

Figure 2

The angular velocity of the locator (Figure 2):

$$
\omega(t)=\frac{\mathrm{d} \alpha}{\mathrm{d} t}=\frac{v}{h} \cdot \frac{1}{1+\frac{v^{2} t^{2}}{h^{2}}}
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_271f369abba9e95c27d9g-1.jpg?height=358&width=458&top_left_y=1813&top_left_x=817)

Figure 3

Eliminating \( t \) using (1) gives us \( \omega \) as a function of \( \alpha \) (Figure 3):

$$
\omega(\alpha)=\frac{v}{h} \cos ^{2} \alpha
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_271f369abba9e95c27d9g-2.jpg?height=447&width=588&top_left_y=129&top_left_x=758)

Figure 4

The angular acceleration is the time derivative of the angular velocity (Figure 4):

$$
\beta(t)=\frac{\mathrm{d} \omega}{\mathrm{d} t}=-2\left(\frac{v}{h}\right)^{2} \frac{v t}{h} \cdot \frac{1}{1+\left(\frac{v^{2} t^{2}}{h^{2}}\right)^{2}}
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_271f369abba9e95c27d9g-2.jpg?height=521&width=726&top_left_y=978&top_left_x=686)

Using (1) (Figure 5):

$$
\beta(\alpha)=-2\left(\frac{v}{h}\right)^{2} \frac{\operatorname{tg} \alpha}{\left(1+\operatorname{tg}^{2} \alpha\right)^{2}}=-2\left(\frac{v}{h}\right)^{2} \cdot \sin \alpha \cos ^{3} \alpha
$$

The angular acceleration has a local maximum where its derivative is zero:

$$
\frac{\mathrm{d} \beta}{\mathrm{d} \alpha}=-2\left(\frac{v}{h}\right)^{2} \cos ^{2} \alpha\left(\cos ^{2} \alpha-3 \sin ^{2} \alpha\right)
$$

The interesting zeros of this are at \( \alpha = \pm 90^{\circ} \) and \( \alpha = \pm 30^{\circ} \). Among these, the maximum occurs at \( \alpha = -30^{\circ} \), where the derivative changes from positive to negative.

Schmidt József (Esztergom, Dobó K. Gymnasium, 4th grade)
On a straight country road, a car travels at a constant speed $v$. Its motion is observed by a radar located at a distance $h$ from the road. Determine the angular velocity and angular acceleration of the radar's rotating antenna as a function of time and angular displacement. Plot the functions and determine the angular displacement at which the angular acceleration is maximal.