When entering a damaged section of the highway, each car in a convoy reduces its speed from v_1 to v_2. What should be the distance between the cars so that they do not collide? The length of each car is l.

Solutions of Savchenko Problems in Physics

Aliaksandr Melnichenka
October 2023

    <h3 id="back-link"><a href="/#1.2">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $1.2.5.$ When entering a damaged section of the highway, each car in a convoy reduces its speed from $v_1$ to $v_2$. What should be the distance between the cars so that they do not collide? The length of each car is $l$.
    </p>

    <h3>Solution</h3>
    <p>
        <p>

Let's move to the frame of reference associated with the $1$th car at the moment when it enters the damaged area and starts moving with velocity $v_2$. The second car will move with a velocity equal to the velocity difference
$$v_\text{from}=v_1-v_2$$

Время, которое проедет второй автомобиль, в системе отсчета «остановленного» первого автомобиля равно

$$t=\frac{L}{v_1–v_2},\\, (1)$$

где $L$ - дистанция между автомобилями.

За время $t$ первый автомобиль с измененной скоростью $v_2$ должен проехать минимум расстояние равное длине автомобиля

$$t=\frac{l}{v_2} \\, (2)$$

Приравняем $(1)$ и $(2)$

$$\frac{L}{v_1–v_2}=\frac{l}{v_2}$$

Откуда искомая дистанция между автомобилями, при которой они еще не столкнутся равна

$$\fbox{$L \geq \frac{v_1–v_2}{v_2} \cdot l$}$$

    </p>

    <h4>Answer</h4>
    <p>
        $$L=\frac{v_1–v_2}{v_2} \cdot l$$
    </p>


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