Word Problem...

[Kent]

At exactly what time between 2 and 3 will the minute hand pass the hour hand?

Sounds pretty simple but it's not.

I have two ideas about what the answer is, before I post them I would like to see what you all come up with.

Thanks for your time and effort,
-Kent

 

[Famine]

Answer 1: It cannot. This is a subtle version of Xeno's Paradox. As the minute hand approaches where the hour hand was, the hour hand has moved on a little further. As the minute hand approaches where the hour hand just was then, the hour hand has again moved on a little further.

Answer 2: 2.11pm and 11 seconds.

 

[Dave A]

With regards to answer one, it does pass the hour hand, every hour at a different point during the hour, while the hour hand is moving away from the minute hand, the mnute hand is moving at a faster rate than the hour hand, so it eventually catches up. If they were moving at the same rate then they'd never meet. The distance each hand moves is the same ammount each time they move, so I don't see where zeno's paradox could fit.

Ignor that, I see it now.

 

[Famine]

With regards to answer one, it does pass the hour hand, every hour at a different point during the hour, while the hour hand is moving away from the minute hand, the mnute hand is moving at a faster rate than the hour hand, so it eventually catches up. If they were moving at the same rate then they'd never meet. The distance each hand moves is the same ammount each time they move, so I don't see where zeno's paradox could fit.

It's simple.

The hour hand is a 2. The minute hand is at 12.

When the minute hand reaches 2, the hour hand has reached 2.2. So the minute hand must now travel to 2.2.

When the minute hand reaches 2.2, the hour hand has reached 2.21. So, the minute hand must now travel to 2.21.

When the minute hand reaches 2.21, the hour hand has reached 2.211. So, the minute hand must now travel to 2.211.

When the minute hand reaches 2.211, the hour hand has reached 2.2111. So, the minute hand must now travel to 2.2111.

When the minute hand reaches 2.2111, the hour hand has reached 2.2111. So, the minute hand must now travel to 2.21111.

And so on, literally, ad infinitum. This is Xeno's Paradox - Achilles can never overtake a tortoise, since every time he reaches where the tortoise has been, the tortoise has moved on a little further.

It is patently rat turd, as my second answer shows.

 

[GilesGuthrie]

It is patently rat turd, as my second answer shows.

As every analogue clock in the whole world shows.

Even the s*** ones made in Taiwan.

 

[Dave A]

Zeno's paradox says you can split a set distance into an infinite number of sections but not time, and using his own logic, you can split it up in a similar way. So using the same logic he's used for splitting up a distance you can give yourself infinite time to cover that distance. It is as Famine put it, rat turd.

 

[daan]

What does a warrior princess know about paradoxes?

or whatever the plural of paradox is....

 

[Zardoz]

What does a warrior princess know about paradoxes?...[/i]

He's getting completely out of hand. This site needs a "pun filter" or something...

 

[Jpec07]

It's simple.

The hour hand is a 2. The minute hand is at 12.

When the minute hand reaches 2, the hour hand has reached 2.2. So the minute hand must now travel to 2.2.

When the minute hand reaches 2.2, the hour hand has reached 2.21. So, the minute hand must now travel to 2.21.

When the minute hand reaches 2.21, the hour hand has reached 2.211. So, the minute hand must now travel to 2.211.

When the minute hand reaches 2.211, the hour hand has reached 2.2111. So, the minute hand must now travel to 2.2111.

When the minute hand reaches 2.2111, the hour hand has reached 2.2111. So, the minute hand must now travel to 2.21111.

And so on, literally, ad infinitum. This is Xeno's Paradox - Achilles can never overtake a tortoise, since every time he reaches where the tortoise has been, the tortoise has moved on a little further.

And yet, the minute hand does pass the hour hand, therefore, at some point in time (maybe in between second-ticks), the two hands are exactly in line. All your math can't negate the physics of it: the hour hand makes one twelfth of a rotation in an hour while the minute-hand makes a whole rotation: they have to cross.

As for the problem, I use a digital watch, so you lose.

 

[Famine]

And yet, the minute hand does pass the hour hand, therefore, at some point in time (maybe in between second-ticks), the two hands are exactly in line. All your math can't negate the physics of it

It's got bog all to do with me. Hence why it's called "Xeno's Paradox" - and it isn't "math", it's philosophy.

 

[Kent]

Interesting take on the question...

Here's what I've been thinking.

It takes exactly 12 minutes for the Hour hand to move from the "10-minute" mark to the "11-minute" mark (since there are 4 minute marks between each number, and the number itself is the fifth mark) (5 into 60 = 12).

So, with that in mind... The minute hand has to pass the hour hand before 2:11.

That is because the hour hand will not cover the space between the 10 and 11 minute marks until 12 minutes have passed.

My estamate: 2:10.52 - 2:10.53

Remember, at 11 minutes into the hour, the minute hand has made it to the 11 minute mark which the hour hand reaches only after 12 minutes.

So at 10 minutes into the hour, the hour hand is 10/12 of the way from the 10 minute mark to the 11 minute mark.

At that same time, the minute hand has made it to the "2" or 10 minute mark.

Then it becomes a matter of seconds.
With in the next 60 seconds the minute hand will pass the hour hand.

Since the hour hand is 10/12ths of the way through the minute-mark, the second hand will have to move for atleast 10/12ths of a minute before the minute hand will move 10/12ths of the way through the minute.

However, at 11/12ths of the way through the minute the minute-hand will have moved past the hour hand (since the hour-hand doesn't make it to 11/12ths until 11 minutes have gone by.

So... Once again, I think it is...
After 2:10.50
Before: 2:10.55

What do you all think?

 

[Dave A]

I think I'll have to read that again when I don't have a hang over , Ive never really thought about that, but I'll have to do some calculations when I get home to check it over.

 

[Randymcchickenf]

ok, this is what i thought / did

the hour hand moves at 1/12 or a rotation an hour, the minute moves at 1 rotation an hour.
The hour hand starts at 2 and the minute starts at 12.

this can be written as two equations,

Y=X and Y= (X/12)+(2/12)

so X=(1/12)X +2/12

so (11/12)X = 2/12

so x = 2/11 = 0.181818...

.18 of an hour is 10mins 48 seconds, so the hands pass at 2.10.48

hope thats clear enough, heres a pic anyway to try and explain it.

 

[Dave A]

You've just saved me doing that myself, I wouldn't have done a graph too though so 👍.

 

[Randymcchickenf]

haha, well great minds think alike...

 

[niky]

Another paradox of infinities... just like that damn airplane on the conveyor...

 

[Kent]

I see one problem with 2:10.48

At 48 seconds into a minute, the minute hand has not yet moved 10/12ths of the way through the minute mark (from the 10-minute mark to the 11-minute mark).

However, the hour hand will have already passed the 10/12ths mark because 10 minutes have already gone by.



Does that put us at square one again?

I think one of the biggest problems in this is that the best way I have found to deal with the problem is in fractions and fractions do not lend themselves to mathmatics with ease.

Like I said though...

10/12ths of a minute is 50 seconds.
At 2:10 the hour hand has moved to 10/12ths of the minute mark.

To me that means it must be after 50 seconds of the minute.
Like-wise, it must be before 55 seconds of the minute because 55 seconds into the minute will leave the minute hand 11/12ths of the way through the minute mark.

So...

2:10.50 to 2:10.55

Somewhere between those 2. 👍

 

[Kylehnat]

I calculate 2:10.54.6 (real close to 2:10.55)

Using rotational velocities, we can figure this out.

The minute hand travels all the way around in one hour, or 360 degrees (if we treat the clock as a 360 degree circle) or 6.283 radians/hour. The hour hand travels 1/12 of the way around or 6.283/12=0.524 radians/hour. The minute hand starts at a position of zero degrees (zero radians), and the hour hand starts at a position of 2/12*6.283 radians or 1.047 radians.

At any given time, the position of a hand will be its initial position + velocity*time elapsed.

Position as function of time (position in radians, time in hours) is:

minute hand = 6.283t + 0
hour hand = 0.524t + 1.047

When these two equations are equal, the two hands are at the same position. Solving for t gives t = 0.182 hours, or 10.91 minutes (past 2 o'clock). Translating this into a time gives 2:10.54.6 (if i didn't mess up in the conversion).

 

[Dave A]

I agree with Kent.

 

[Randymcchickenf]

does the minute hand tick?....lol

EDIT:

I see one problem with 2:10.48

At 48 seconds into a minute, the minute hand has not yet moved 10/12ths of the way through the minute mark (from the 10-minute mark to the 11-minute mark).

However, the hour hand will have already passed the 10/12ths mark because 10 minutes have already gone by.



Does that put us at square one again?

ok...well it seems your correct. now im just confused...have to retry this...

@ kylehnat - answer looks write, but you lost me in the radians per hour thing....anyone care to educate me upon this?

 

[Randymcchickenf]

i have a new answer, and it IS correct.

using the same theory as i did before, write the 2/12 of an hour as 10mins, and you get

X=X/12 + 10
so 11X/12 = 10
x = 120/11 = 10.90909090...
so the answer is 10mins and 54.5454...seconds

Im not entirely sure why writing 2/12 of an hour makes such a difference from 10mins, as they are the same...maybe i was mixing up my minutes and hours...well thats enough for me...back to not using my brain

 

[Dave A]

@ kylehnat - answer looks write, but you lost me in the radians per hour thing....anyone care to educate me upon this?

Radians is a form of measurement that works a little different to degrees, 360 degrees is equal to 6.283 radians.

 

[Kylehnat]

and 6.283 is equal to 2*pi (no coincidence since we're talking about circles).

 

[Randymcchickenf]

i got the radians, but im used to expressing them in pi rad, so 2pi rad = 360, thats why i was confused. lol. Ok, cheers for that

 

[Kent]

My dad had originally given me this "2:10.545454(repeating)"

Thanks for confirming his work. 👍

My guess work with fractions came pretty close though.
Later.