Clock Hands Collinearity Explained

This problem involves calculating the time lapse until clock hands become collinear again. Collinear means the hands form a straight line, which occurs when they are either overlapping (0° angle) or directly opposite (180° angle).

Relative Speed Calculation

First, determine the relative speed between the minute hand and the hour hand.

• Minute hand speed ($v_m$): $6^{\circ}$ per minute.
• Hour hand speed ($v_h$): $0.5^{\circ}$ per minute.
• Relative speed ($v_{rel}$): $v_m - v_h = 6^{\circ}/\text{min} - 0.5^{\circ}/\text{min} = 5.5^{\circ}/\text{min}$.

Time Lapse for Collinearity

The question states the hands are initially "along the same line". This can mean they are overlapping (0° apart) or opposite (180° apart).

We need the time until they are *next* collinear.

• Case 1: Starting Overlapped (0°). The next collinear position is when they are opposite (180° apart). The time required is the angle difference divided by the relative speed: $ \text{Time} = \frac{180^{\circ}}{5.5^{\circ}/\text{min}} = \frac{1800}{55} = \frac{360}{11} \text{ minutes} $ $ \frac{360}{11} \approx 32.73 \text{ minutes} $
• Case 2: Starting Opposite (180°). The next collinear position is when they overlap again (a relative gain of 360°). The time required is: $ \text{Time} = \frac{360^{\circ}}{5.5^{\circ}/\text{min}} = \frac{3600}{55} = \frac{720}{11} \text{ minutes} $ $ \frac{720}{11} \approx 65.45 \text{ minutes} $

Conclusion

The calculated time lapses are approximately 32.73 minutes and 65.45 minutes.

Comparing these values to the given options:

• 32.73 minutes is closest to 33 minutes.
• 65.45 minutes is closest to 63 minutes.

Since 33 minutes is an option and the likely intended answer based on typical clock problems of this nature, the closest time lapse is approximately 33 minutes.