What is the required speed to average 60 MPH for a car trip with varying speeds?

[lrandf]

Please someone help me with these brain teasers, Thanks! I have try to answer these problem, but I don't know whether they are right or wrong. PLEASE HELP!

A hill is one mile from the bottom to the top and then one mile from the top to thebottom. A person drives a car from the bottom to the top going 30 MPH. How fast does the person have to drive the car down the hill inorder to average 60 MPH fro the entire trip?

My answer is 90MPH

There are four volumes of Shakespeare on the shelf. The pages of each volue are exactly two incles thick. The covers are each one-sixth of an inch thick. A bookworm started eating at page 1 of Volume I and ate straight through to the last page of Volume IV. What distance did the worm cover?

My answer is 9 1/3

Ann

 

[Gokul43201]

Q1) How is average velocity defined ? What is the formula for it ? How do you calculate each of the components in this formula - assume the speed during the second part of the trip is v, and solve for it from the equation for the average velocity.

Q2) This depends on how the books are arranged : I II III IV or IV III II I

The first way is more likely, and the answer there would be much smaller than your guess. Actually stack 4 books the correct way, and do this again.

 

[gerben]

I do not get the first question it just seems to be impossible: if it would take no time to go down hill you would have traveled 2 miles in 1/30 hours (the time it took to travel up hill), this would give the average velocity of 60MPH , but this is of course impossible... if it does take some time the average velocity will be lower...

I am very curious to Gokul's solution, for me writing it down in a formula does not make it any clearer, it is the same thing, I mean it shows exactly the same impossibility...I guess there is some catch I do not see??

 

[VantagePoint72]

First problem:
The problem with saying the speed going down is 90 and then saying ave. speed = (90 + 30) / 2 is that your assuming that the amount of time the car was traveling 30mph and the amount of time it was traveling at 90mph were equal, which isn't the case since the trip in which the car is traveling at 90 will be less time than the trip during which the car is traveling at 30, and therefore the speed 90mph will have a smaller contribution to the overall average. To solve this problem, you have to think in terms of percentage of the entire trip. Unfortunately, I don't have time to give this problem enough thought myself, but try solving it keeping in mind the fact that the trip up the hill and the trip down the hill will be different times so it won't be a simple ave = x + y / 2

Second problem:
According to your wording, the worm doesn't go through the first and last cover. Therefore, his distance is 4(2) + (6/6) = 9.

 

[Gokul43201]

I do not get the first question it just seems to be impossible: if it would take no time to go down hill you would have traveled 2 miles in 1/30 hours (the time it took to travel up hill), this would give the average velocity of 60MPH , but this is of course impossible... if it does take some time the average velocity will be lower...

I am very curious to Gokul's solution, for me writing it down in a formula does not make it any clearer, it is the same thing, I mean it shows exactly the same impossibility...I guess there is some catch I do not see??

You're perfectly correct. If you solve for t2, using v(avg) = 2d/(t1+t2), you will get t2 = 0. This requires an infinite downhill velocity, which as you mentioned, is impossible.

I was only trying to describe the correct method to approach this kind of problem because, as LastOne has mentioned, many people tend to simply take the arithmetic mean.

 

[gerben]

You're perfectly correct. If you solve for t2, using v(avg) = 2d/(t1+t2), you will get t2 = 0. This requires an infinite downhill velocity, which as you mentioned, is impossible.

I was only trying to describe the correct method to approach this kind of problem because, as LastOne has mentioned, many people tend to simply take the arithmetic mean.

Yes ok thanks Gokul, I see, there really was no catch... the question was simply impossible to answer, the "answer" should be: There is no possible velocity that could make the average velocity of the car be 60 MPH.

Although you have to learn that "average velocity" means "the distance traversed divided by the time it took to traverse this distance" I sympathize with you (Irandf) that people are giving you such questions to learn it. I always hated questions like this, and still do, I mean they ask for a speed of a car and there simply is no speed of a car that could answer the question...

It is probably some kind of mathematics guy who asked you this question, they often do not care about any physical possibility of the answer to their problems...

Moreover, the answer: Infinite velocity (whatever that may be) is only valid for those who accept that:
x = infinity
is an solution for:
x = 1/0
which is not intuitive for anybody, and also in mathematics the equation x = 1/0 is said to be not defined, meaning it is not defined what x is is this case...

 

[VantagePoint72]

Perhaps since we're dealing with a real world example, it would help to say that "as one's velocity approaches infinity, the average speed approaches 60mph. At ave. = 60mph, the function is undefined." Not that it changes anything, it just helps with visualizing the situation since infinity has more meaning as far as velocity is concerned than "undefined" does.

 

[Sariaht]

l-...ll...ll...ll...-l
...I...II...III...IV...

9 inch