Expert: Josh - 11/19/2007

Question
QUESTION: At what time between 2:00 o'clock and 3:00 o'clock, to the nearest second, do the hour and minute hands of a clock make a straight angle?


ANSWER: Jennifer,

Maths problems can be solved in many ways. One way of looking at this is that the hour hand (H) and minute (M) hand move at different rates.

Part I:
Measuring time in seconds, writing rH and rM for the rate at which the hour and minute hands rotate around the clock, we can work out the rates as follows:

- One revolution around a circle is equivalent to 360 degrees
- 1 hour = 60 minutes = 3600 seconds

The hour hand moves from 2 to 3 in an hour. Thus, it completes 1/12 x 360 degrees in an hour. This is equivalent to (360/12) degrees for every 3600 seconds.

=> Rate of movement of hour hand: rH=30/3600=1/120 degree/second

The minute hand covers 1/60 x 360 degrees in 60 seconds

=> Rate of movement of minute hand: rM=6/60=1/10 degree/second

Part II: Starting positions of the hour (H) and minute (M) hands.

Let "t" be the time elapsed after 2pm. The position of H at any time "t" is given by the initial position of H, plus  movement given by the product of time and the rate at which H moves around the circle. i.e.,
================================
pH(t) = pH(0) + rH*t
where t: time elapsed after 2pm
     pH(t): position of hour hand at time t
     pH(0): initial position of hour hand at time t=0
     rH: rate at which hour hand moves around circle [deg/sec]
================================
Similarly, the position of the minute hand is governed by the same physical principle.
pM(t) = pM(0) + rM*t
where t: time elapsed after 2pm
     pM(t): position of minute hand at time t
     pM(0): initial position of minute hand at time t=0
     rM: rate at which minute hand moves around circle [deg/sec]
================================

Part III: Angle separating the two hands are given by
|pH(t)-pM(t)|. And we want the difference to be 180 degrees in order to form a straight line.

The bar here means we are taking the magnitude only, ignoring the sign. e.g., |-180| is still regarded as a difference of 180 degrees, it doesn't matter whether H leads M, or M leads H. In fact, we can remove this ambiguity (hence, also the vertical bars) if we look more closely at the situation. Since we start at 2:00, and the movement of the minute hand is much faster than the hour hand, the former will surpass the latter. The moment when the two hands form a straight line, the minute hand is bound to be in front of the hour hand which has not yet reach the number 3.

The straight line condition is described by:
rM(t)-rH(t)=180 [degree] and the initial position of the H and M hands at 2:00 are rH(0)=(2/12)*360 and rM(0)=0, respectively.

Solving rM(t)-rH(t)=180 for time t, using the physics equations established earlier, we want

pM(0) + rM*t - [pH(0) + rH*t] = 180
....substituting the initial values pM(0)=0, pH(0)=60 [deg] and rates of movement rM=1/10 and rH=1/120
0 + t/10 - [60 + t/120] = 180
0 + t/10 - t/120 = 240
(1/10 - 1/120)*t = 240
t = 240/[1/10 - 1/120] = 2618.1818....

This is the elapsed time in seconds since 2pm.

May I ask you to have a go at the remaining task, converting this to a time in hh:mm:ss? I would be happy to clarify anything if you're not sure. I want you to gain something in the process of answering this question.


---------- FOLLOW-UP ----------

QUESTION: I don't understand how you convert the answer into hour minutes and seconds??

Answer
Hi Jennifer,

We just need to remember there are 60 seconds in a minute; and 60 minutes (60 x 60 = 3600 seconds) in an hour.

Since t=2618.181818... is less than 3600, the number of hours to add on is 0. The time on the clock is 2:mm:ss.

We use the division algorithm to find the number of minutes in 2618.181818 seconds. Dividing 2618.181818 [sec] by 60 [sec/min], we get 43.63636363...minutes. Rounding this down to the nearest number, we get 43. So, the time on the clock is 2:43:ss.

The fractional part 0.63636363...is still measured in minutes. To convert this to seconds, we divide 0.63636363 [minute] by (1/60)[minute/sec]. This yield approximately 38.18 seconds. We can round this to the nearest integer to produce the final answer -- 2:43:38.

My teachers always advise me to check my answer. It's one thing for the answer to be off by a small margin, quite another for it to be blown out of proportion.

To check that this number makes sense, observe that 43.63 minutes is almost 45/60 or 3/4 of an hour. Thus, we expect the minute hand to be 3/4 of the way around the circle on the clock face. In fact it is slightly less. For the hour hand, we expect it to have rotated about 3/4 of the way between 2 and 3. A quick sketch should convince us that the two hands are roughly 180 degrees apart.

More precisely, we can use the equations we developed to verify the result. Only for a homework problem, if you have time on your sleeves.

From pH(t) = pH(0) + rH*t, substituting t=2618.1818..., rate rH=1/120 and initial position pH(0)=60, we get pH(2618.1818) = 60+2618.1818/120 = 81.81 [degree].

For the minute hand, its position is given by pM(t) = pM(0) + rM*t. Substituting t=2618.1818..., rate rH=1/10 and initial position pH(0)=0, we get pH(2618.1818) = 2618.1818/10 = 261.81 [degree].

What is the separation between the two hands?