We need to find what the time is when the hands of the clock are at 180 degree to each other for the first time after 12:00. Let this be after x minutes. Now we know that the minute hand moves (360/60)*x degrees in x minutes. The hour hand moves (360/12*60)*x degrees in a minute. Now when the two hands are at 180 degree to each other it means that (360/60)*x - (360/12*60)*x = 180
=> 2x/60 - 2x/12*60 = 1
=> x (1/ 30 – 1/360) = 1
=> x = (1/ 30 – 1/360) ^-1
=> x = 32.727 min
Therefore the two hands are at 180 degree to each other at 32.72 minutes past 12:00.
Every hour the minute hand of the clock moves one full circle or 360 degree. During the same period the hour hand moves by 1/12th of a complete circle, or 30 degrees.
Thus every 1 hour after 12:00 the net angle between the hour hand and minute hand increases by:
360 - 30 = 330 degrees
The time taken for the angle between hour and minute hand to increase from by 180 degrees from 0 to 180 degrees is given by:
Time for the angle to increase by 180 degrees = 180*(1/330)
= 6/11 hours.
= (6/11)*60 = 32.7272 minutes = 32 minutes and 44 seconds
Time after 12:00 when hands of clock are at 180 degrees to each others = 12:32:44
We know that the hour hand rotates one full round in 12 hours or 12*60 = 720 minutes. So the hour hand covers 360 degrees in 720 minutes. Therefore the hour hand moves with a speed of 1/2 degrre per minute.
The minute hand moves 360 degree every hour. So the speed of the minute hand per minute is 360/60 = 5 degree per minute.
Therefore difference in the angle of hour hand and minute hand per minute = speed of minute hand/minute - speed of hour hand per minute = 5-1/2 deg/min = 4.5 degree per minute.
Therefore it takes 180 deree/ 4.5 = 40 mintes for the angle to be 180 degree for the first time between the hour hand and the minute hand.