Problem Analysis

The question asks for the number of times the second-hand and the minute-hand of a clock coincide (cross each other) within the specific time interval from 12:05:00 to 12:55:00.

Understanding Clock Hand Speeds

To find when the hands cross, we need their speeds:

• Minute Hand Speed ($v_m$): Completes 360 degrees in 60 minutes. $v_m = \frac{360^\circ}{60 \text{ min}} = \frac{360^\circ}{3600 \text{ s}} = 0.1^\circ/\text{s}$
• Second Hand Speed ($v_s$): Completes 360 degrees in 60 seconds. $v_s = \frac{360^\circ}{60 \text{ s}} = 6^\circ/\text{s}$

Calculating Relative Speed and Coincidence Interval

The second hand overtakes the minute hand because it moves faster. We calculate the relative speed ($v_{rel}$) at which the second hand gains on the minute hand:

$v_{rel} = v_s - v_m = 6^\circ/\text{s} - 0.1^\circ/\text{s} = 5.9^\circ/\text{s}$

The hands coincide when the second hand gains a full 360 degrees relative to the minute hand. The time interval ($T_{coincidence}$) between consecutive coincidences is:

$T_{coincidence} = \frac{360^\circ}{v_{rel}} = \frac{360^\circ}{5.9^\circ/\text{s}} = \frac{3600}{59} \text{ seconds}$

Determining Coincidences within the Interval

We consider time $t$ in seconds starting from 12:00:00. Coincidences occur at times $t_k = k \times T_{coincidence} = k \times \frac{3600}{59}$ seconds, where $k$ is a non-negative integer.

First, convert the given time interval to seconds past 12:00:00:

• Start Time: 12:05:00 = $5 \times 60 = 300$ seconds.
• End Time: 12:55:00 = $55 \times 60 = 3300$ seconds.

We need to find the number of integers $k$ such that the coincidence time $t_k$ falls within the interval [300, 3300] seconds.

$300 \le k \times \frac{3600}{59} \le 3300$

Solve for $k$:

$k \ge \frac{300 \times 59}{3600} \implies k \ge \frac{59}{12} \approx 4.9167$

$k \le \frac{3300 \times 59}{3600} \implies k \le \frac{11 \times 59}{12} = \frac{649}{12} \approx 54.0833$

The possible integer values for $k$ are $5, 6, 7, \dots, 54$.

Final Count

To find the total number of coincidences, count the number of possible integer values for $k$:

Number of coincidences = (Last value of $k$) - (First value of $k$) + 1

Number of coincidences = $54 - 5 + 1 = 50$

Therefore, the second-hand and minute-hand cross each other 50 times between 12:05:00 and 12:55:00.