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)
- Ajay walks at a speed of 4 km/hr. He doubles his speed after reaching exactly half way. He walks for 12 hours in all. What is the total distance travelled by him ?
- A train travels from Karnal to Kurukshetra at a speed of $90$ km/hr and returns at a speed of $92$ km/hr. If it takes $182$ hours for total travel, then find the distance between Karnal to Kurukshetra. (In Km)
- A car covers a distance from Hayathnagar to LB nagar at the speed of 64 km/hr and covers the distance from LB nagar to Hayathnagar at the speed of 56 km/hr. What is the average speed (rounded off to nearest integer) of the car?
- A delivery van travels 156 km in 3.5 hours at a constant speed. What is its speed in km/hr (rounded off to the nearest whole number)?
- Priyanka covers 800 m distance in 6 minutes. Her speed in km/hr is:
