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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]


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