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-   -   How fast is the dog? (https://www.mersenneforum.org/showthread.php?t=11664)

 Andi47 2009-03-31 11:06

How fast is the dog?

A dog decides to run 500 km. It has got an empty cane tied on its tail, which hits the ground exactly once every second with a loud "clonk!" sound.

The dog starts its run with the first clonk it hears, with a velocity of 2 m/s. Afterwards it doubles its velocity instantly (i.e. with infinite acceleration) every time when it hears a clonk from the cane.

How fast is the dog when it reaches its destination?

 akruppa 2009-03-31 11:16

Assuming Newtonian mechanics, [spoiler]262144m/s[/spoiler].

Alex

 Andi47 2009-03-31 11:24

[QUOTE=akruppa;167416]Assuming Newtonian mechanics, [spoiler]262144m/s[/spoiler].

Alex[/QUOTE]

Yes, we assume Newtonian mechanics, but you are far off.

 Mini-Geek 2009-03-31 12:04

[quote=Andi47;167417]Yes, we assume Newtonian mechanics, but you are far off.[/quote]
Are you sure? I get the same answer. I made a script for my calculator to find it by summing, but here's a more advanced way to describe the logic:
[spoiler]At x seconds, (where x is an integer) the dog was running at 2^x m/s, accelerates to 2^(x+1) m/s, and has traveled 2^(x+1)-2 m (2^(x+1)-2 is equivalent to the sum of 2^1 ... 2^x and I might've used that instead, but 1. TeX-formatted equations, like I'd need to use the Sigma function, don't spoilerize, and 2. I don't already know the TeX format for Sigma). 2^(x+1)-2 first becomes 500000 or higher during the 18th second, and 2^18=262144, so the dog was running at 262,144 m/s when he passed the 500 km mark. On a side note, I think it took the dog approx. 17.907 seconds to reach the 500 km mark.[/spoiler]

 Visu 2009-03-31 12:06

[SPOILER]Umm... 0 m/s.If it has reached its destination it would have stopped...:smile:[/SPOILER]

 Visu 2009-03-31 12:16

Otherwise I have to agree with akruppa and Mini-Geek

 wblipp 2009-03-31 12:31

[spoiler]
512 m/s.

This is the doubling that passes the speed of sound for reasonable temperatures. After that point, the dog never hears another bounce of the cane.

[url]http://hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe.html[/url]
[/spoiler]

 akruppa 2009-03-31 12:35

Clever. I thought there must be some trick to it...

Alex

 Andi47 2009-03-31 12:51

@wblipp - correct!

 Mini-Geek 2009-03-31 12:56

[quote=Andi47;167428]@wblipp - correct![/quote]
[spoiler]I thought it was odd that the question was phrased that way, (about hearing the clonk instead of just that every second he doubled his speed) but you never specified the speed of sound in the atmosphere the dog is running, so how were we to know that this dog that can accelerate instantly and run 500 km at once happened to be in our atmosphere under standard conditions, and not in some hypothetical medium that transfers sound faster than 262144 m/s?[/spoiler]

 Andi47 2009-03-31 13:06

[QUOTE=Mini-Geek;167430][spoiler]I thought it was odd that the question was phrased that way, (about hearing the clonk instead of just that every second he doubled his speed) but you never specified the speed of sound in the atmosphere the dog is running, so how were we to know that this dog that can accelerate instantly and run 500 km at once happened to be in our atmosphere under standard conditions, and not in some hypothetical medium that transfers sound faster than 262144 m/s?[/spoiler][/QUOTE]

[spoiler]Hmmm... Mayby I should have stated something like "the dog runs along the Interstate 95 from Miami to Jacksonville, assuming that there is no traffic on the highway". That would have implied, that the run happens on earth with "reasonable" temperatures and thus a speed of sound of ~333 m/s, without giving away hints by words like "atmosphere", "temperature", etc.[/spoiler]

 Mr. P-1 2009-03-31 15:05

[QUOTE=Andi47;167417]Yes, we assume Newtonian mechanics[/QUOTE]

I would have thought the infinite acceleration stipulated in the question rather foreclosed the applicability of Newtonian mechanics.

 Orgasmic Troll 2009-03-31 15:45

Puzzles like this should be banned, and violators should be woken up by wet-willies for the rest of their life.

The puzzle is set up already violating physical laws, but we're expected to pick out whichever physical law the puzzlemaker arbitrarily decides to keep in. Stupid stupid stupid.

You might as well ask "What color am I thinking of?"

Oughtn't a dog capable of infinite acceleration also be capable of hearing sound transmitted through its body from tail to ears, and, so, hearing the vibrations transmitted thusly from cane to tail through body to ears? What's the speed of sound through dog flesh, skeleton and tendons?

[quote=Orgasmic Troll;167447]The puzzle is set up already violating physical laws, but we're expected to pick out whichever physical law the puzzlemaker arbitrarily decides to keep in.[/quote]Yes, that is an implicit, unstated part of the puzzle, and part of the fun of discussing it.

[quote]You might as well ask "What color am I thinking of?"[/quote]Well ... knowing you ... hmmm ... actually, I don't know you that well, but I can guess anyway ... [spoiler] [I]ultraviolet[/I]. [/spoiler]

Oh, wait -- you're hypothesizing that the [I]reader[/I] is asking someone else, "What color am I thinking of?" Then, the usual answer is [spoiler]"seven" (though a few Doug Adams fans might venture "forty-two")[/spoiler].

 wblipp 2009-03-31 18:16

[QUOTE=cheesehead;167448]Oughtn't a dog capable of infinite acceleration also be capable of hearing sound transmitted through its body from tail to ears, and, so, hearing the vibrations transmitted thusly from cane to tail through body to ears?[/QUOTE]

I worried about that point, too. But I decided that although the dog could hear the cane hitting by this method, the sound would be a thud rather than a clonk much as our own voices sound deeper to us when speaking than on recordings. Hence I concluded the puzzle was carefully worded to avoid this possibility.

 R.D. Silverman 2009-03-31 18:44

[QUOTE=wblipp;167467]I worried about that point, too. But I decided that although the dog could hear the cane hitting by this method, the sound would be a thud rather than a clonk much as our own voices sound deeper to us when speaking than on recordings. Hence I concluded the puzzle was carefully worded to avoid this possibility.[/QUOTE]

Except that whatever the sound IS, it happens [b]behind[/b]
him. Once he is moving faster than Mach 1 he never hears the sound
again.........

 alpertron 2009-03-31 19:36

Well, the sound can pass through the dog's body because the cane is tied to it so the limit is not Mach 1, but at these speeds there are louder noises than the "clonk" the original poster states.

 Uncwilly 2009-03-31 22:45

[QUOTE=wblipp;167426][spoiler]512 m/s.[/spoiler][/QUOTE]
I dispute that, the [B][U]only[/U][/B] assumtion that we are asked to make is the acceleration. Therefore there are two potential answers.

1) [spoiler]~20 m/s, the max speed of any pooch. Greyhounds can burst out to ~45 mph, thus that is the max speed.[/spoiler]
or
2) [spoiler]No speed, because the can not sustain anywhere near that speed over the course of 500km. Therefore, any dog that tries to run that far at top speed will die of exhaustion.[/spoiler]

 alpertron 2009-04-01 01:48

3) [spoiler] The heat generated will rise the temperature of the dog above 100 degrees Celsius, with obvious problems with the water inside the body.[/spoiler]

 retina 2009-04-01 02:35

[QUOTE=Uncwilly;167495]... the [B][U]only[/U][/B] assumtion that we are asked to make is the acceleration.[/QUOTE]Ermm, I think you mean the only physical law we are being asked to suspend is acceleration limits (and maybe inertia and momentum). We are being expected to implicitly assume a multitude of things, like the metre and second being the SI standard, that the dog actually can make such decisions as to run 500km for some unknown reason etc. etc. etc.

I think a more appropriate answer is:

4) [spoiler]The dog never makes to the destination because it gets distracted by hunger, thirst, a rabbit it saw or a bitch in heat. And perhaps multiple of those distractions in succession.[/spoiler]

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