Do you have mastery over the clock and angles?

Have you ever stared at a clock and wondered exactly how many times the hands hit a specific angle? It sounds like a simple counting task, but because both hands are constantly moving at different speeds, it’s actually a challenge of Relative Velocity.

Clock and Angles

Let’s break down the solution for the interval between 4:00 a.m. and 12:00 noon.

The Secret Formula: Relative Speed

To solve any clock problem, you first need to know how fast the hands move in degrees per minute:

• Minute Hand: Moves 360∘ in 60 minutes = 6∘ per minute.
• Hour Hand: Moves 30∘ (one-twelfth of a circle) in 60 minutes = 0.5∘ per minute.

Because they are both moving the same way, the minute hand “gains” on the hour hand at a rate of:

6∘−0.5∘=5.5∘ per minute

How Often Does the Angle Occur?

The hands form a 150∘ angle in two different “positions”:

1. When the minute hand is 150∘ behind the hour hand.
2. When the minute hand is 150∘ ahead of the hour hand.

In every full “lap” the minute hand makes around the hour hand (which takes about 65 minutes), these two positions will each happen once.

Mapping Out the 8-Hour Window

The time from 4:00 a.m. to 12:00 noon is exactly 8 hours. Let’s look at the “zones” of time to see where the angles fall.

Zone 1: The Start (4:00 a.m. — 5:00 a.m.)

At exactly 4:00, the hands are 120∘ apart (4×30∘).

• Because they start at 120∘, the minute hand has already passed the “behind” position for this cycle.
• It will only hit the “ahead” position once (this happens at approximately 4:49 a.m.).
• Count: 1 time

Zone 2: The Middle (5:00 a.m. — 11:00 a.m.)

During these 6 hours, the hands go through their full range of motion. Since each “hour” of a clock’s cycle generally allows for both the “ahead” and “behind” positions to occur:

• 6 hours × 2 occurrences per hour = 12.
• Count: 12 times

Zone 3: The End (11:00 a.m. — 12:00 noon)

At 11:00, the hands are 30∘ apart (with the hour hand ahead).

• The minute hand will catch up, pass the hour hand, and reach the 150∘ ahead position once.
• However, it won’t be able to reach the 150∘ “behind” position before the clock strikes 12:00.
• Count: 1 time

The Final Verdict

Adding up all our zones:

• 4:00–5:00: 1 time
• 5:00–11:00: 12 times
• 11:00–12:00: 1 time

Total: 14 times.

Summary Table for Quick Reference