Task solved by StrgAltEntf

# Task 5 - 060935 

On the circle $k$, point $A$ moves with a uniform speed $v_{1}$ and point $B$ moves with a uniform speed $v_{2}$, where $v 1 \neq v_{2}$.

If both points move in the same direction (e.g., clockwise), point $A$ overtakes point $B$ every 56 minutes. If both points move in opposite directions, they meet each other every 8 minutes. In this case, their distance measured along the circumference increases or decreases by $14 \mathrm{~m}$ every $24 \mathrm{~s}$.

a) How long is the circumference of the circle?

b) What are the speeds $v_{1}$ and $v_{2}$ (in $\mathrm{m} / \mathrm{min}$)?
a) According to the conditions of the problem, $v_{1}>v_{2}$. When moving in different directions, the relative velocity of $B$ with respect to $A$ is obtained by adding their velocities. According to the problem, with this relative velocity, $14 \mathrm{~m}$ are covered in $24 \mathrm{~s}$, and in $8 \mathrm{~min}$ thus

$$
\frac{14 \cdot 60 \cdot 8}{24} \mathrm{~m}=280 \mathrm{~m}
$$

This is the length of the circumference.

b) As stated,

$$
v_{1}+v_{2}=\frac{14 \mathrm{~m}}{24 \mathrm{~s}}=35 \frac{\mathrm{m}}{\mathrm{min}}
$$

If the points move in the same direction, the relative velocity of $B$ with respect to $A$ is obtained by subtracting the smaller velocity $v_{2}$ from $v_{1}$. According to the problem and the result from a), this is thus

$$
v_{1}-v_{2}=\frac{280 \mathrm{~m}}{56 \mathrm{~min}}=5 \frac{\mathrm{m}}{\mathrm{min}}
$$

The sought velocities are therefore $v_{1}=20 \frac{\mathrm{m}}{\mathrm{min}}$ and $v_{2}=15 \frac{\mathrm{m}}{\mathrm{min}}$.

Taken from [5]