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| <h3 id="back-link"><a href="../../#1.2">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.2">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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 $l$. | | $1.2.5.$ 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> | | </p> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <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 | | 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 |
| $$v_\text{from}=v_1-v_2$$ | | $$v_\text{from}=v_1-v_2$$ |
| </p> | | </p> |
| <p> | | <p> |
| Время, которое проедет второй автомобиль, в системе отсчета «остановленного» первого автомобиля равно | | Время, которое проедет второй автомобиль, в системе отсчета «остановленного» первого автомобиля равно |
| </p> | | </p> |
| <p> | | <p> |
| <center>$$t=\frac{L}{v_1–v_2},\, (1)$$</center> | | <center>$$t=\frac{L}{v_1–v_2},\, (1)$$</center> |
| </p> | | </p> |
| <p> | | <p> |
| где $L$ - дистанция между автомобилями. | | где $L$ - дистанция между автомобилями. |
| </p> | | </p> |
| <p> | | <p> |
| За время $t$ первый автомобиль с измененной скоростью $v_2$ должен проехать минимум расстояние равное длине автомобиля | | За время $t$ первый автомобиль с измененной скоростью $v_2$ должен проехать минимум расстояние равное длине автомобиля |
| </p> | | </p> |
| <p> | | <p> |
| <center>$$t=\frac{l}{v_2} \, (2)$$</center> | | <center>$$t=\frac{l}{v_2} \, (2)$$</center> |
| </p> | | </p> |
| <p> | | <p> |
| Приравняем $(1)$ и $(2)$ | | Приравняем $(1)$ и $(2)$ |
| </p> | | </p> |
| <p> | | <p> |
| <center>$$\frac{L}{v_1–v_2}=\frac{l}{v_2}$$</center> | | <center>$$\frac{L}{v_1–v_2}=\frac{l}{v_2}$$</center> |
| </p> | | </p> |
| <p> | | <p> |
| Откуда искомая дистанция между автомобилями, при которой они еще не столкнутся равна | | Откуда искомая дистанция между автомобилями, при которой они еще не столкнутся равна |
| </p> | | </p> |
| <p> | | <p> |
| <center>$$\fbox{$L \geq \frac{v_1–v_2}{v_2} \cdot l$}$$</center> | | <center>$$\fbox{$L \geq \frac{v_1–v_2}{v_2} \cdot l$}$$</center> |
| </p> | | </p> |
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| </p> | | </p> |
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| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$L=\frac{v_1–v_2}{v_2} \cdot l$$ | | $$L=\frac{v_1–v_2}{v_2} \cdot l$$ |
| </p> | | </p> |
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