Rope Winding Speed Calculation

The problem asks for the speed at which a boy approaches a pole while winding a rope of length $l$ around it. The radius of the pole is $r$, and each round takes $10\text{ s}$.

Length of Rope Wound Per Round

When the boy completes one full revolution ($2\pi$ radians) around the pole, the amount of rope that gets wound onto the pole is equal to the circumference of the pole.

Length wound in one round = $ 2\pi r $

Rate of Rope Winding

The time taken for one round is given as $10\text{ s}$. The rate at which the rope winds onto the pole is the length wound divided by the time taken.

Rate of winding = $ \frac{2\pi r}{10\text{ s}} $

Simplifying this expression gives:

Rate of winding = $ \frac{\pi r}{5} \text{ units/s} $

Speed of Approach to Pole

As the rope winds around the pole, the length of the *unwound* portion of the rope decreases. This decrease in unwound length directly corresponds to the boy moving closer to the pole. Therefore, the speed at which the boy approaches the pole is equal to the rate at which the rope winds up.

Speed of approach = Rate of winding = $ \frac{\pi r}{5} $