The Hands of a Clock Coincide at Noon. How Many Minutes Must Elapse Before They Coincide Again?
35. A regular clock has an 60 minutes and infinitesimal hand. At 12 midnight the easily are exactly aligned. When is the next time they will exactly align or overlap? How many times a day volition they overlap?
I thought of this puzzle as a manner of explaining another puzzle. But it's quite a good one so we can encompass it itself...
(The 'Hands in a Straight Line' trouble is further down the page.) First thing is the answer is not i:05 which might exist your first idea. Why? As we dealt with in puzzle #v by the fourth dimension the minute mitt gets to v past one the hour hand will have moved slightly past 1 o'clock. The hour hand after all does non expect at 1 o'clock for an hour just rather it moves to 2 o'clock over the class of the adjacent hour. The first overlap will be a footling subsequently five by i.
The first approach is probably the most obvious; it's the most mechanically intuitive. We'll use a variable 't' the time in hours. We know the speeds of the hands by definition the Minute Hand will move i full rotation or 360° per hour. The Hr Paw makes a full rotation in 12 hours and will therefore movement at 30° per hr. At our first overlap just after five by ane the Minute Manus volition have done one full rotation plus the bit nosotros are interested in. The Hr Mitt will accept done just a part rotation of 't' times it'southward speed.
Hour Hand = Minute Hand 30t = 360t - 360 t = 12(t - ane) 11t = 12 t = 12/11 = 1.090909.. hours = 65.454545454.. minutes = 1h 5min 27.272727..seconds
The next method i would like to suggest is that we realise that the concept of 360 degrees is an artificial structure. And non necessary to solve the problem. We can consider the speeds only in term of revolutions per hr with the Minute and Hour Easily moving at 1 and 1/12 revolutions per hr respectively. The first line of our equation and so becomes:
60 minutes Hand = Minute Hand t/12 = t - 1
Probably the about elegant solution is to realise that the overlap will happen 11 times in a 12 hr flow and so the respond is merely i/11th of 12 hours. Information technology's hard to divine this. But information technology makes sense. Simply as we reasoned our commencement overlap volition exist at just afterwards i:05 we can reason that our last volition exist just before 10:55. Our overlaps volition exist at just after i:05, a bit more after two:10, even more later on three:fifteen and then on (actual times requite at the cease) but since our last is a bit before x:55 at that place volition exist no overlap whilst the Hr Mitt is indicating xi. A Very Scientific Approach below also gives us a way to realise this.
I found the whole problem a lot easier to visualise past playing with the flash based clock below. You can drag the easily with your mouse.
Click here to view wink content...
Flash Clock kindly provided past MathsIsFun
A Very Scientific Arroyo
The terminal method is very physics. I want to expect at the concept of reference frames, the idea has many applications in physics, information technology's essential for example in full general relativity or when nosotros consider something like how fast is the light traveling away from the head lights of a fast moving car. But it also has some much more mundane applications. Simply we redefine our co-ordinate system so that information technology's stationary with respect to one of our bodies. An example of this would be if we have 2 cars moving at different speeds, lets say thirty and 35mph. The faster car is a mile backside, when will it overtake? Nosotros can solve the equation of motility 30t+1=35t, only what we might practice intuitively is move to the frame of reference of the pb motorcar. In this reference frame the lead auto is stationary and the car behind is a mile away approaching at 5mph. That's substantially a 1 dimensional problem and a fleck too like shooting fish in a barrel to solve. But if yous are considering say the intercept courses of ships in 2 dimensions information technology becomes more useful.
Nosotros will make our reference frame that of the Hour Hand. (Either would work.) The problem will exist from the perspective of someone stood on the Hour Manus with no other visual cues; all he tin see is the minute hand. In the external reference frame the Minute Hand moved with speed (or angular velocity, since nosotros are now speaking physics) 1 revolutions per hr and the Hour Hand with ane/12th. To move into the frame of reference of the 60 minutes Mitt nosotros subtract information technology's speed from that of the Minute Hand to requite the Minute Mitt's speed in the Hour Manus'southward reference frame. As in one - 1/12th. In our new frame of reference the Minute Hand moves at an angular velocity of 11/12ths revolutions per hr. Coincidence or overlap occurs every time the Minute Mitt makes a complete revolution. The question of how many revolutions something moving at xi/12ths revolutions per hour makes in a 12 60 minutes catamenia is petty.
If we have by now proven that there are 11 overlaps in a 12 hour period, conspicuously there are 22 overlaps in a 24 hour period.
Easily course a direct line
The first approach is probably the virtually obvious; information technology's the nearly mechanically intuitive. We'll use a variable 't' the time in hours. We know the speeds of the hands by definition the Minute Mitt will movement 1 full rotation or 360° per hr. The Hr Hand makes a full rotation in 12 hours and will therefore motion at 30° per 60 minutes. The first straight line afterwards midnight will be simply after one-half an hour. We are looking for a point where the position of the hr paw plus 180 degrees is equal to the position of the minute manus.
Hour Hand + 180 = Minute Hand 30t + 180 = 360t t + 6 = 12t 6 = 11t t = half-dozen/xi = .545454.. hours = 32.727272.. minutes = 32min 43.636363..seconds
Total List of Overlap Times
Both AM and PM
01:05:27
02:ten:55
03:16:22
04:21:49
05:27:16
06:32:44
07:38:11
08:43:38
09:49:05
10:54:33
12:00:00
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