How fast do I have to go?

This is the kind of question you'd get on an SAT test. The true theoretical answer is yes it's possible to average 60 mph, but only if you can travel the 2nd mile at infinite speed. But here's the formal mathematical answer:

let V = velocity, D = distance and T = time

by definition we know that V = D / T

So in this word problem we want to consider two velocities and their average. So let's call the velocity around the 1st mile V1 and the velocity around the 2nd mile V2.

Let D1 be the distance around the 1st mile and D2 be the distance around the second mile. Of course this means that D1 = D2.

Let T1 be the time to travel the 1st mile at a velocity of V1. Let T2 be the time to travel the 2nd mile at a velocity V2.

Finally, let: A = be the average velocity around the total distance. So we get:

A = (D1 + D2)/(T1 + T2) = (total distance) / (total overall time)

but we know that:
V1 = D1/T1 so that T1 = D1/V1

similarly we can show that T2 = D2/V2

So if I go back to:

A = (D1 + D2)/(T1 + T2) and substitute for T1 and T2 we get:

A = (D1 + D2)/ (D1/V1 + D2/V2)

But we already know that D2 = D1 so we can say

A = 2*D1 / (D1/V1 + D1/V2)

After simplifying the above we get:

A = 2*V1*V2 / (V1 + V2)

So the above is a general formula for describing the average velocity if you go around any track twice going the first time at a speed of V1 and the 2nd time at V2.

But to find the answer to your question we want to solve for V2 knowing that V1 is 30 mph and our goal is to get A = 60 mph.

You can show that:

V2 = (A*V1) / (2*V1 - A)

So if I plug in the numbers I get:

V2 = (60 mph)*(30mph) / [(2 * 30 mph) - 60 mph]
V2 = 1800 / (60 - 60) = 1800 / 0 = infinity

However, as a practical matter if you could travel the speed of light on the 2nd lap you'd probably be fast enough that we wouldn't notice that it took you any time to travel the 2nd lap and so you'd be close enough to a 60 mph average.

Now here's the next and more interesting math problem that I haven't solved yet, but is kind of at the crux of how you can maximize your gas mileage in a Prius using the pulse and glide driving technique. If you drive the first mile averaging a terrible 5 mpg because your going full throttle up a steep hill in a Prius and then coast the next mile on an almost flat surface at 100 mpg, what's your average mileage. Intuitively you might be inclined so say it's somewhere just above 50 mpg. But, I'm not entirely sure if that holds up mathematically and if you need to know the speed you are traveling while hitting these mileages. Incidentally, the answer to this 2nd math problem of average mileage sort of guides how I drive. In pulse and glide you realize that you'll get poor mileage if you siimply try to go easy on the gas. In pulse and glide you find out that you'll fair much better if you start off hard on the gas to reach your desired coasting speed, and then abruptly lift off the gas enough so that you coast so as to allow you gas engine to shut off completely.

 

Gosh darnit! I thought the original post was pretty clear and straightforward, and the answer ("It's impossible" or "You'd have to travel infinitely fast") was just as obvious. I'm amazed it generated such a long discussion.

I'm also kind of surprised, given the whimsical nature of folks here, that nobody said, It's only possible if you have the EV button.

 

90mph

oval or not, it matters not, it's 1 mile at 30mph average
if mile number 2 averages out to be 90mph, then 90+30=120/2=60

answers to the contrary are simply wrong.

 

Intuition is off:

1 mile / 5mpg = .2 gallons
1 mile / 100mpg = .01 gallon
Avg mpg = 2 miles/.21 gallons = 9.5 mpg

Even if you used no gas for your 2nd mile, you would have used .2 gallons for your first mile and average 2 miles/.2 gallons = 10mpg.

 

Problem is you were driving at 30mph three times longer than you were driving at 90mph.

So it's 30mph for the 1st 40 seconds, 30mph for the 2nd 40 seconds, 30mph for the 3rd 40 seconds and 90mph for the last 40 seconds.

30 + 30 + 30 + 90 = 180 / 4 = 45 mph.

Glad it's all cleared up now.

 

OH MY GOODNESS.

this thread screams "GEEK!" :lol:

good thing i have no recollection of calculus or i'd really annoy you all. [whew]

 

And, these bunch of geeks (me included, and I just got a binary watch for X'mas gift) don't seem to agree with one another on a simple problem...

 

Time ain't the issue here.

Averaging 30mph over 1 mile is averaging 30mph over 1 mile. No one mentioned anything specific about time, acceleration issues, etc.

The answer to the problem given the way it IS WRITTEN is what myself and several others came up with. Now, either imnotacrook transcribed it incorrectly, or the answer he says is right is, in fact, wrong.

Lots of folks here looking into it TOO deeply, coming up with shiat that ain't there...

 

I been out drivin my Prius in 30 and 90MPH cirlces for the last hour. 60 Minutes, 30@30, 30@90, average of who cares, headache, bald tires, 3 beers. Oh and 2 gallons of gas.

OU

I'm callin MythBusters!

 

No time limit. Only distance Limit 2 miles
1 mile at 30
1 mile at 90
Average for 2 miles of distance 60.

Just like your fuel economy. 30mpg for a mile 90mpg for a mile = 60mpg average.

 

Did you READ the problem?

The first mile is NOT at 30mph but an AVERAGE of 30mph.

 

This actually made sense to me. And I realized that yes, this would make your average lap time 60mph.

However, the original question, said how fast would you have to go to average 60mph over the whole 2 miles, which would be impossible.

I'm not even going to believe this will clear anything up.

 

How about this:

How Slow would I have to go?

I was driving the Prius around a 1 mile oval, my average speed for the first mile was 90 MPH. How slow do I have to go on the second lap, to average 60 MPH for the entire 2 miles?? And what is you reasoning?




Answer Stop!

 

Okay -- would you mind answering a few questions, though?

1. How much time does the first lap (1 mile at 30 mph) take?
2. How much time does the second lap (1 mile at 90 mph) take?
3. How much total time was taken by the first and second laps combined?
4. How much distance was covered by the two one-mile laps?
5. What do you get when you divide the total distance covered by the total amount of time spent travelling?

6. Moving away from the original question -- imagine that you travel one mile at 1mph. This would take you an hour, right? If you travelled a second mile at 59mph, would your average speed be ((59 + 1) / 2) = 30 mph? Wouldn't that mean the 2-mile trip would take a grand total of four minutes? What happened to the hour you spent on the first mile?

 

It's an AVERAGE, it can be a whole array of speeds as long as when divided by a number results in 30

For example,

it could be 60 and 0....

or 120, 0, and 0, still equates to an AVERAGE of 30mph right?

 

lol, GREAT thread!

I think whomever wrote that problem, really needs to understand the concept of "0"....

again, you could average 30 with 120,0,0, at 120mph, how long do you think it would take to drive a mile?

Sure as hell not 2 minutes.

Or even 30, 60, 0, 30 whatever as long as the AVERAGE equates to 30


I should know... :lol:

 

I think the concept that's causing difficulty here is "average speed". This is a term that comes up in math classes a lot and is useful for things like estimating how long a trip is likely to take. If you take H hours to travel M miles, then your average speed is M/H mph, and a hypothetical car travelling at a constant rate of M/H mph would take exactly as long to complete its trip as you took to complete yours. "Average speed" isn't something that the author of this math problem made up, and it doesn't just mean to take the averages of several speeds.