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} $
Important Questions from Speed Time & Distance (Notes)
A and B have to travel from place P to place Q following the same route in their respective cars. A drives at $60$ kmph while B drives at $80$ kmph. Find the time taken by B to reach place Q if A takes $12$ hrs.
On a straight road, a bus is $60$ km ahead of a car running in the same direction. After $3$ hours, the car is $90$ km ahead of the bus. If the speed of the bus is $45$ km/h, then what is the speed of the car (in km/h)?
A train running at the speed of $90$ kmph crosses a $250$ m long platform in $26$ seconds. What is the length of the train (in m)?
A car covers 4 successive stretches of 3 km each at speed of 10 kmph, 20 kmph, 30 kmph and 60 kmph respectively. The average speed of the car for the entire journey is:
- A car travels a total distance L. It travels half the distance with speed $v_1$ and the other half with speed $v_2$. The average speed of the car is :
