Clock Hands Crossing Problem
The question asks for the number of times the minute-hand and the second-hand of a clock coincide between 09:15:00 AM and 09:45:00 AM.
Minute and Second Hand Speeds
First, determine the angular speed of each hand in degrees per second:
- Second-hand: Completes 360 degrees in 60 seconds. Speed = $\frac{360^\circ}{60 \text{ s}} = 6^\circ/\text{s}$.
- Minute-hand: Completes 360 degrees in 60 minutes (3600 seconds). Speed = $\frac{360^\circ}{3600 \text{ s}} = 0.1^\circ/\text{s}$.
Calculating Hand Crossing Frequency
The minute and second hands cross when the faster second-hand overtakes the minute-hand. This requires the second-hand to gain a full 360 degrees relative to the minute-hand.
Calculate the relative speed:
Relative Speed = Speed(Second-hand) - Speed(Minute-hand)
Relative Speed = $6^\circ/\text{s} - 0.1^\circ/\text{s} = 5.9^\circ/\text{s}$.
The time between consecutive crossings is:
Time per Crossing = $\frac{360^\circ}{5.9^\circ/\text{s}} \approx 61.02$ seconds.
This implies they cross approximately once every minute. To be precise, within any 60-second interval [hh:mm:00, hh:mm+1:00), the second hand starts at 0 degrees and the minute hand is at some angle $\alpha$. They are guaranteed to cross exactly once within that interval.
Analyzing the 09:15 to 09:45 Interval
The specified time interval is from 09:15:00 AM to 09:45:00 AM.
The total duration is 30 minutes.
The interval spans the following 1-minute periods where crossings occur:
- [09:15:00, 09:16:00)
- [09:16:00, 09:17:00)
- ...
- [09:44:00, 09:45:00)
The number of these 1-minute periods is $45 - 15 = 30$.
Since there is exactly one crossing per minute, there are 30 crossings in total.
Final Count of Crossings
The minute-hand and second-hand cross 30 times between 09:15:00 AM and 09:45:00 AM.
Important Questions from Clocks
- Rohit goes to a restaurant for lunch at about $1 \text{ PM}$. When he enters the restaurant, he notices that the hour and minute hands on the wall clock are exactly coinciding. After about an hour, when he leaves the restaurant, he notices that the clock hands are again exactly coinciding. How much time (in minutes) did Rohit spend at the restaurant?
- On a given day, how many times will the second-hand and the minute-hand of a clock cross each other during the clock time 12:05:00 hours to 12:55:00 hours?
- A worker noticed that the hour hand on the factory clock had moved by 225 degrees during her stay at the factory. For how long did she stay in the factory?
- In a 12-hour clock that runs correctly, how many times do the second, minute,and hour hands of the clock coincide, in a 12-hour duration from 3 PM in a dayto 3 AM the next day?
- It is quarter past three in your watch. The angle between the hour hand and the minute hand is ______
