General Cycling Discussion - Quiz Time! - Average Speed Problem

Here is a little quiz for everyone. I will give the question now and the answer later on today or tomorrow. Post your answers and let's see who gets it right! :D

Question:

You have a 1 mile loop racing track. Your goal is to average 60 mph for two laps. You do the first lap at 30 mph. How fast must you go on the second lap in order to average 60 mph?
:confused:

90

Not very difficult ;)

-bs

To have an average speed of 60 mph for 2 miles, those 2 miles would have to be completed in 2 minutes. Since the first lap at 30 mph would take 2 minutes, it would be impossible to have an average of 60 mph for the two laps, no matter how fast you went on the second.

Originally posted by sscyco
To have an average speed of 60 mph for 2 miles, those 2 miles would have to be completed in 2 minutes. Since the first lap at 30 mph would take 2 minutes, it would be impossible to have an average of 60 mph for the two laps, no matter how fast you went on the second.

I think he's talking about average speed for the laps, in which he would need to go 90 mph as mentioned previously.

originally posted by Stor Mand
I think he's talking about average speed for the laps, in which he would need to go 90 mph as mentioned previously.

Stor Mand, figure that out on paper - it doesn’t calculate - the question he asks is a trick question, it can't be done. I think.

Hmmm, interesting question.

Going by speeds alone.
30mph on the first lap
90mph on the second lap
equals 60mph doing a simple average.

But the above doesn't take into consideration distance and time, let's look at the time/speed/distance measurements
Lap1 - 30mph for 1 mile takes 2 minutes.
Lap2 - 90mph for 1 mile takes 45 seconds.
Total time = 2:45 = for 2 miles 45mph

Essentially you can't go 2 miles in 2 minutes if you average 30mph for the first mile, which means in 2 miles you couldn't average 60mph if you went 30mph for the first mile no matter how fast you go for the second mile.

Andrew

Originally posted by sscyco
Stor Mand, figure that out on paper - it doesn’t calculate - the question he asks is a trick question, it can't be done. I think.

He didn't mention time. If it is a time thing, then it is not possible. I was going strictly by the information given.
If the time were to be 2 minutes, obviously time would be up in the 1st lap.

:beer:

originally posted by Stor Mand
He didn't mention time. If it is a time thing, then it is not possible. I was going strictly by the information given.

Yes he did - mph = miles per HOUR

Originally posted by Stor Mand


He didn't mention time. If it is a time thing, then it is not possible. I was going strictly by the information given.
If the time were to be 2 minutes, obviously time would be up in the 1st lap.

:beer:

Doesnt' matter about mentioning time, when you ask a speed for distance question time has to be part of it.

The original question;

You have a 1 mile loop racing track. Your goal is to average 60 mph for two laps. You do the first lap at 30 mph. How fast must you go on the second lap in order to average 60 mph?


The question asks a question based on distance (2 miles), and average speed (target 60mph). In order to go 2 miles at an average speed of 60mph means you have to complete those 2 miles in 2 minutes. So if you go the first mile at 30mph, there is no way to average 60mph over 2 miles.

This is just a trick question, where the most obvious answer isn't the right one.

It's kind of like the following question.
3 guys need to spend a night at the hotel, the hotel charged them $30 for the room, which they split $10 for each guy. A little while later the hotel manager found a mistake and overcharged them $5, so he gives the bellboy the $5 to return to the three guys. Now in a quandry, the bellboy doesn't know how to split up the $5 into three equal portions, so he keeps $2 for himself and gives the three guys each $1.

So now the three guys each spent $9 for the room, for a total of $27, and the bellboy kept $2.....total of all that money is $29....but the original room price was $30. Where is the missing dollar?

Andrew

Its a verbage thing -
30-5=25+3+2=30

Originally posted by sscyco
Its a verbage thing -
30-5=25+3+2=30

Like asking what is thirty, divided by one half, plus ten?

But to answer the original question - if you strictly look for the average speed over the 2 laps, the average will be 60 if the second lap was 90.
If you look for the average speed by the time it took for the 2 laps, the average would be 48, as stated above.
In a way there are 2 correct answers to this.

:beer:

So, based on the reasoning others have used--that the 30 mph uses up the two minutes implicitly allotted--my answer is that he would have to go infinitely fast in the second lap.

Now, I am quite sure that I cannot go infinitely fast, and I would expect most of those on this forum cannot go that fast either. Some would wager that Lance Armstrong can go that fast (I have my doubts).

But, the answer is not "impossible," its "infinitely fast."

Cheers,
Jamie

Here is the correct answer! Good job to those who got it correct.


No, it is *NOT* 90 mph. If you do the first lap at 30 mph, it takes you 2 minutes for one lap. An average of 60 mph is 2 laps in 2 minutes. Thus, it is *IMPOSSIBLE* to average 60 mph if you do the first lap at 30 mph (even if you could travel at the speed of light--which you can't, because you have mass--you'd still be averaging slightly over 60 mph.

The place where people's intuition breaks down is that the basis for finding average speed is time, not distance. So, while it would be true that one minute at 30 mph and one minute at 90 mph *does* give you a 60 mph average, it doesn't work if you do one mile at 30 and one mile at 90. The reason is that you spend two minutes doing 30 mph but only 40 seconds doing 90 mph (which comes out to 45 mph).

Originally posted by Captain Crunch
So, while it would be true that one minute at 30 mph and one minute at 90 mph *does* give you a 60 mph average, it doesn't work if you do one mile at 30 and one mile at 90.
The reason is that you spend two minutes doing 30 mph but only 40 seconds doing 90 mph (which comes out to 45 mph).

So, since you state that the above is true, doesn't it mean that there are actually 2 possible answers? The only issue is that one is more correct than the other.
:beer:

Originally posted by Stor Mand


So, since you state that the above is true, doesn't it mean that there are actually 2 possible answers? The only issue is that one is more correct than the other.
:beer:

Nope, there is only one correct answer. He wants to go two miles and average 60mph for those 2 miles. If he does the first mile at 30mph there is no possible way to average 60mph over the 2 miles.

In fact the speed you have to go isn't even "infinate", it's just not possible. Once you've done the first mile you would have to instantly finish the 2nd mile, so no matter how fast you go you would never average 60.0mph, you may average 60.0000000000000000000000000000000000000000000000000000000000000000000000001 mph, but not 60mph even.

Andrew

Actually! No, there is only one correct answer for this question as it was asked.

But to answer the original question - if you strictly look for the average speed over the 2 laps,

SPEED IS time and distance. Miles per HOUR, Kilometers per SECOND, whatever, you cannot state a SPEED without having a distance. Speed is distance per unit of time.

uh stor mand... there's only one correct answer... and it's already been given. math isn't open to interpretation (at least not the algebra and calculus involved in this problem). the fundamental thing to remember is the difference between instantaneous and average speeds. best to think of it as a hare and tortise problem. maybe even exaggerate the problem to make the solution a little more clear to those who aren't as mathematically inclined. say a "tortise"" is running along at 60 miles an hour on a 2 mile course LOL. Now say a lazy old hare starts at the same exact time, kinda being lazy for the first mile, just movin along at 30 miles an hour thinking... man, i'm so much faster than that tortise... i'll just slack off now, and really give my best effort in the last mile. well see, the hare can only slack off for so long, until there will be a point where no matter how fast he goes... even going the speed of light, the tortise will have reached the end of the two mile course while the hare is still back there somewhere on the course. Actually, the point of which I speak is the one that Cap'n Crunch gave... the 1 mile mark. Going 60 miles an hour, the tortise would have already reached the end by the time the hare got to the 1 mile mark with his 30 mph pace, thus making it impossible for him to catch up

Ok... now for the more mathematically inclined...
remember that fundamentally, average velocity is just distance travelled over time. say that by some miracle the guy was able to go a zillion miles an hour on the second lap. that would mean 2 miles covered in 2 minutes and 0.000000000001 seconds. That will give you an average speed of 59.9999999 mph. This is what we call an asymptote. As your speed approaches infinity on the second lap, you will get VERY close to 60 mph, but never actually reach it. now if the guy had gone 30 mph for say... 0.8 mile, then he'd actually have a chance of making it to the 2 mile mark with an avg speed of 60 mph. but he "missed the boat" so to speak.

Originally posted by deliriou5
uh stor mand... there's only one correct answer... and it's already been given. math isn't open to interpretation (at least not the algebra and calculus involved in this problem).


Yes, there is only one correct but there are 2 true answers, as Captain Crunch has already stated below. :rolleyes:


Originally posted by Captain Crunch
snip snip ...
So, while it would be true that one minute at 30 mph and one minute at 90 mph *does* give you a 60 mph average snip snip

And yes, he did say it was true. I do understand that being true does not make it correct.

And here as stated by ajay213:


Going by speeds alone.
30mph on the first lap
90mph on the second lap
equals 60mph doing a simple average.

It's a simple average. Not the correct answer, but it is a true answer.

And your wrong about math not being open to interpretation (statistics) ... anyone with some math knowledge can manipulate the numbers to get the answer that is needed. :p
:beer:

average velocity is just distance travelled over time

Thanks for the nice explanation.

Just to keep things on the up and up, velocity is speed in a given direction., or "distance traveled in unit time in a given direction."

At least that is what it says in my friendly Webster's, and what we teach our sixth graders in science.