Posted: 3/21/2002 11:45:09 AM EDT
[Last Edit: 3/21/2002 11:51:26 AM EDT by IMHO]
You are driving by car to a particular destination, and our only assumption is that you are free to drive at any speed you choose  no traffic jams or anything like that. For the first half of the journey (i.e.half the distance) you drive at 20 miles per hour. You then realise that this is all taking much too long, and that you are going to be late. You therefore decide that you will increase your speed so that your overall average speed for the whole journey will be 40 miles per hour. How fast do you have to drive for the remaining part of your journey in order for your average speed for the whole journey to be 40 miles per hour?



60 mph. Assuming the switch is exactly halfway thru (or near enough it makes no odds) increasing speed to 60 mph will yield an average speed of 40mph.
(60+20)/2 = 40. Average of two speeds.
What do I win?
FFZ



FreeFireZone:
Sorry, that's not correct.
I'm afraid that I can award only praise and respect for the right answer.



The answer is you cannot drive fast enough for the remaining distance to raise your average speed to 40.



FreeFireZone, you'd be correct if he said half the time instead of half the distance. It is impossible to finish the whole journey with an average speed of 40 mph if you went the first half of the distance at 20 mph. This is because if 0.5*x/t=20mph, then x/t=40mph (where x is the whole distance and t is time). You have to finish the remaining 0.5x without taking any more time. Impossible.



Hawkman:
Well Done!
I hold you in high regard.
With immense respect,
IMHO
[And also to BostonTeaParty for the rationale]
Continue to be impressed with the breadth of knowledge here. 60 mph is the intuitive choice. It was also my first thought. Thanks for responding.



Thankyouthankyouthankyou. All those years of grad school finally paid off![:D]



So in all honesty, the answer would be infinity.


Please consider the paper industry before not printing this message.

The answer is that it's impossible for you to drive the second half of your trip fast enough to average 40 mph for the whole.
Suppose, for instance, that your trip was 40 miles long. If you drove 20 mph to the halfway point, one hour would have already elapsed, so any time spent covering the remaining distance would make the denominator in the distance/time equation too big.
This assumes that you can't cheat by turning around, speeding back to your starting point, and heading back again to double the distance covered. [;)]



This problem is just begging for a stochastic (calculus) analysis. Limits, integration, etc...
Basically the limit is 2x your previous speedin this case 40mph/ph. You can never exceed this limit due to the distance qualificationyou have a finite distance remaining to be traveled that is equal to the distance already traveled. You could concievable get up to 39.99999999999999999 mph/ph for the entire trip, but you would never break 40mph/ph as the average speed for the distance traveled over the time traveled.
Quantum physicists would argue that time would bend as you approach the speed of light, and that time would seem to stand still, but I digress....[rolleyes]
I love math![bounce]



What if you were in an airplane, and your airspeed was 20 mph, but your ground speed was actually higher? Theroretically, you could make it.
[=(_)=]


Please consider the paper industry before not printing this message.

Figures. To paraphrase DeForest Kelley  "I'm a mechanic, not a mathematician, dammit!"
FFZ (but I'm one HELL of a mechanic!)



BYU:
I think that physicists interested in kinematics would argue that miles/hour/hour is a unit of acceleration.
(Not trying to be picky. . . just want to support your love of math)



Originally Posted By IMHO:
BYU:
I think that physicists interested in kinematics would argue that miles/hour/hour is a unit of acceleration.
(Not trying to be picky. . . just want to support your love of math)
View Quote 


What if I go 2^(aleph_0) miles per hour?



Do you mean "c"?



Incorrect. This fails to take into account the Theory of Relativity by which a car can effectively travel at the speed of light and time will stop.
Edited to tell BYU: No fair, I had to change a diaper.


My politics are the faith, principles and intent of the Founding Fathers of this Great Nation, formerly known as the "melting pot".
I support the people and State of Israel. Teach PeaceLive War. Teach WarLive Peace. 
Axel:
I guess that you are one of the quantum physicists that BYU was referring to. A fine life pursuit.
"This fails to take into account the Theory of Relativity by which a car can effectively travel at the speed of light. . . "
The point of the brain teaser, of course, was to challenge intuition within the boundaries of reality. Have never been included in a serious discussion that considered cars capable of travelling at 3e8 m/s.



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