Two Laps

[SamBones]

You are going to run exactly two laps of a track. You can run the first lap at any speed you want as long as it's a constant speed. How fast do you need to run the second lap for your total average speed to be twice the speed of the first lap?

Apologies if I didn't describe it clearly. I'll clarify the problem if necessary.

 

[DaveInIowa]

Spoiler

1) It's not possible because you've used up all the time to run the 2 laps at twice the speed during your initial lap, or 2) You run at the speed of light where no time elapses during the 2nd lap.

 

[karluk]

[hide] I calculate that it's impossible to run fast enough on the second lap to get your average speed up to twice the average speed of the first lap. For example, if you run the first lap in a minute (thus averaging one lap per minute), you would have to complete the second lap in no time at all in order for your total average speed to be two laps per minute. The same type of calculation shows that you always need to run infinitely fast on the second lap, regardless of your speed on the first lap.

A couple of observations: First, in real life it's impossible to measure speed with infinite precision, so it might be of some interest to calculate just how fast you would have to run in order to get your time on the second lap down to zero, when rounded to the precision of your stop watch. Second, I am not sure why you emphasized that the first lap has to be run at a "constant speed". It seems to me that you would still need to run the second lap instantaneously, regardless of how variable your speed was on the first lap.[/hide]

 

[mintjulep]

Spoiler

One Lap distance = D
Time for first lap = T1
Time for second lap = T2

The average speed of the first lap is: D/T1

The total average speed is: 2D/(T1+T2)

Condition to be met: 2D/(T1+T2)=2D/T1

Simplifying: T1 = T1+T2

T2 must = 0

Therefore the speed of the second lap must be infinity.

 

[kwbMitel]

[Hide] As per my usual, guess and check says infinite speed. My best explanation, the second lap needs to have zero time to be able to double your average speed.[/hide]

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What's most important is that you realise ... There is no spoon.

 

[johnherman]

You need to use warp speed. For instance, let's say it takes an hour to do the first lap. You want your average speed to be half an hour. To get an average of one half hour, you need to average in a lap time of zero.

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[Olaf Doschke]

Spoiler

If the track length is L and lap times are t1 and t2 we need 2L/(t1+t2) = 2 L/t1. which means t2 has to be 0 or the speed for the second round has to be infinite.

Bye, Olaf.

Olaf Doschke Software Engineering

https://www.doschke.name

 

[SamBones]

Heh, not too surprised. All correct answers!

 

[SamBones]

Well, actually, the correct answer is that it's not possible, not that the speed is infinite.

From a purely mathematical standpoint, infinite speed might be possible, but we live in a physical world that does act weird when relative speeds increase. When speed is approaching the speed of light, there's both dilation of both time and space (we'll discount mass for now assuming we have infinite energy available).

So, the greatest speed we can possibly reach for the second lap is the speed of light. I won't do the math, but even close to the speed of light means you will reach the finish line after a non-zero amount of time.