Calculating Average Speed for a Two-Lap Race

[elsmith2]

a multiple choice question from one of my tests:

A race car covers the first lap of a two lap race with an average speed of 90mi/hr. With what speed must the driver average for the second lap so that the average speed for the two laps is 180mi/hr?
A)90mi/hr
B)180mi/hr
C)270mi/hr
D)360mi/hr
E)None of these

I chose C (seemed logical to me, disregarding the fact that race cars don't go 270mi/hr-not the ones that do laps anyways,: (90+270)/2 = 180), but it was wrong. Why? Whats the right answer?

Ed

 

[Math Is Hard]

well, hmm... let's say a lap of a race track is 90 miles long. On the first lap, you are traveling 90 mph so you finish the first lap in an hour (or 60 minutes). The second lap will also be 90 miles long, but let's say you go an average of 270 mph on the last lap. That means you will finish that last lap in 1/3 hour or 20 minutes.
So for 60 minutes you went 90 mph and then for 20 minutes you went 270 mph.
I think that's right, anyway... Does that help explain?

 

[Tide]

The total time to complete the two laps at the new average speed is

\frac {2L}{\bar v} = \frac {2L}{2 v_1} = t_1

I.e. the driver must complete BOTH laps in the same amount of time as he covered the first lap. Therefore ...? :-)

 

[elsmith2]

i worked it out with an arbitrary track length = 2mi (more nascar like than 90mi) and:

90mph(2mi^-1) = 1/45hr
Xmph(2mi^-1) = 2/Xhr

180 = 4/((1/45) + (2/X))

X = 360/(4-4)

so am i to understand that is impossible for the car to avg 180mph (get div by 0 with track length = 90mi too, get it for all track lengths)

ed

 

[dextercioby]

The total time to complete the two laps at the new average speed is

\frac {\bar v}{2L} = \frac {2 v_1}{2L} = t_1

I.e. the driver must complete BOTH laps in the same amount of time as he covered the first lap. Therefore ...? :-)

Mr.ScienceAdvisor+Homework Helper,take a good look at what you've written and tell me if u see something wrong.I believe that u have misswritten a formula taught in the first grades at school.
Since laps are equal wrt to length,different average speeds per lap,mean automatically different times for each lap.Analyzing the formulas written correctly for each lap,you find that the mean speed for the second lap is infinite.
My guess...

 

[rayjohn01]

Must be Japanese

Vave = 2.S / ( t1 +t2 ) = 180 ---------------- 1)

S = t1. v1 = t2 . v2 and v1 = 90

remove t1 and t2 in 1) by substitution 's' cancels , v2 ----> infinity.
Ray

 

[ZapperZ]

I wrote a solution to a similar problem a while back for some "online activities". So rather than regurgitate the whole thing here, I'll just enclose the file. Hopefully it will explain the answer to this problem.

Take note that you should NEVER just simply add the two numbers and divide by two. In statistics, you can do that only if each of those numbers carries the same "weighting" factor. If these are just numbers, then fine. But the "average velocity" has a specific definition. One must always, without fail, consider how things are defined first, before diving head first into tackling a problem.

Zz.

 

[dextercioby]

Thanks Zapper,u just made my point,and not only mine.I'm still surprised by Tide's mistake.He's a science advisor...Unacceptable. Which brings me to a question.If "admin" and "mentor" are comprehendable notions,how does one get to be a "science advisor"??

 

[ZapperZ]

Thanks Zapper,u just made my point,and not only mine.I'm still surprised by Tide's mistake.He's a science advisor...Unacceptable. Which brings me to a question.If "admin" and "mentor" are comprehendable notions,how does one get to be a "science advisor"??

Well, I'm a "Science Advisor" myself. I hate to think that with that title, I am not allowed to make a few mistakes of my own (and which I have!). I don't think the title was ever meant to imply "Perfect Knowledge At All Times".

Zz.

 

[Gokul43201]

Mr.ScienceAdvisor+Homework Helper,take a good look at what you've written and tell me if u see something wrong.I believe that u have misswritten a formula taught in the first grades at school.
Since laps are equal wrt to length,different average speeds per lap,mean automatically different times for each lap.Analyzing the formulas written correctly for each lap,you find that the mean speed for the second lap is infinite.
My guess...

Mr. "shoot-before-you-read"-dextercioby,

Take a good look at what Tide has written. There's absolutely nothing wrong with it, and it only gives the same result that you have arrived at - that it requires an infinitely large speed during the second lap. And not only is Tide's approach consistent with yours (or mine), it is far more elegant.

 

[dextercioby]

Thanks for the nickname,but i'd rather be called "dexter" or "daniel",or anther nickname preferably shorter.
Well,i remember that time is never equal to speed devided by space.Or am i wrong here as well and i may have missunderstood that awkward notation...? I took "t_1" as time,"v" as (mean) velocity and L as distance.So what he's written there should be bull****.
I myself make and will make errors.But at least near my username it's not written "science advisor+homework heper",and the errors I make are not associated with very simple things.That is embarassingly simple things.


To Zapper,i think that title in some cases hides the fact that the whole skyscraper has a sand foundation and someday,out of the blue,it may collapse... But not like a state vector...

 

[Tide]

dex,

Thanks for catching my typo - but you do need to LIGHTEN UP! :-)

 

[Gokul43201]

The only mistake there was writing t instead of 1/t. That's a typo, not a conceptual error.