Find at what time between 8 and 9 O'clock will the hands of...

Find at what time between 8 and 9 O'clock will the hands of a clock be in the same straight line but not together?
(a) $11 \frac{9}{11}$ minutes past 5
(b) $9 \frac{7}{11}$ minutes past 5
(c) $10 \frac{10}{11}$ minutes past 8
(d) None of these

Key Concepts

Relative Motion of Clock Hands

This concept involves understanding that the hour and minute hands move at different constant speeds around the clock. The relative movement between the two hands is what causes them to form specific angles at various times, and analyzing this relative motion allows us to set up equations to determine when the hands will form a particular angle, such as being collinear or in a straight line.

Angular Velocity of Clock Hands

The minute hand and the hour hand each move at a constant rate, measured in degrees per minute. Typically, the minute hand moves at 6 degrees per minute while the hour hand moves at 0.5 degrees per minute. This difference in angular velocity is fundamental when calculating the positions of the hands at any given time and is key to solving problems related to clock angles.

Straight Line Formation (Collinearity)

In the context of clock problems, forming a straight line means that the two clock hands are 180 degrees apart. This condition is different from the hands overlapping and requires setting up the correct equation to represent a 180 degree angle difference. Understanding this property is essential when determining the exact time the hands are in a straight line but not coinciding.

Algebraic Methods for Clock Angle Problems

Solving clock problems typically involves setting up an algebraic equation that represents the angle between the two hands. By expressing the positions of the hands in terms of time variables and using their angular velocities, one can solve for the time at which a specific angular separation (like 180 degrees) occurs. This technique is a critical tool in approaching and solving a range of clock-related mathematical problems.

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