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# Average speed?

11/06/2006 4:31 PM

Here's the question. Suppose you had a distance to travel of 75 miles. For the first mile you travel at 75 mph. When you're 74 miles from your destination, you reduce your speed to 74 mph. When you're 73 miles from your destination, you reduce your speed to 73 mph., and so on, until, when you're 1 mile from your destination you are traveling at 1 mph.

How long does it take to get there?

When I heard this riddle, I assumed that since I was traveling one mile at each speed between 1 and 75, my average speed would be 38 mph, and it would take me about 2 hours.

This is horribly incorrect.

Why? And what is the correct answer?

(AND NO FAIR RESPONDING TO THIS IF YOU HEARD THE PUZZLE ON NPR'S "CAR TALK")

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Associate

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#1

### Re: Average speed?

11/06/2006 6:53 PM

4.901.. hrs or 4 hrs 54 min 4.880.. sec

time = 1/75 + 1/73 + 1/72 + 1/71 ... 1/1

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#2

### Re: Average speed?

11/06/2006 11:23 PM

You will never reach your destination as you speed will approach zero just as your distance to destination approaches zero...forever...

How many times can you divide a line between to points by half? :P

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#3

### Re: Average speed?

11/07/2006 1:28 AM

I think mech_eng's answer is correct. The question does not state the velocity is inversely proportional to the miles left. It is rather discrete with 1 miles of constant speed. Hence, he will eventually make it to the destination.

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#4

### Re: Average speed?

11/07/2006 1:46 AM

Average speed is 15.3018 mph.

A correction on my previous comment: "The question does not state the velocity is proportional to the miles left."

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#5

### Re: Average speed?

11/07/2006 1:49 AM

The summation for step wise speeds appears correct. (4.901hr)

However, if you restate the question so that the speed is always equal to the distance remaining then it is not a sum but an integral that does not converge so you will take forever to reach your destination. Time=integral [1/(75-x)]dx from limits of 0 to 75 =infinite.

As a further example, if it was half mile increments it would be 5.591hrs, 1/4 mile is 12.56, 1/8mile is 27.9 hrs, etc.

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#6

### Re: Average speed?

11/07/2006 2:12 AM

Yes, then;

x=75-75e-t

v=75e-t

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#7

### Re: Average speed?

11/07/2006 2:15 AM

Quote: "Time=integral [1/(75-x)]dx from limits of 0 to 75 =infinite."

Another interesting variant is: start at distance zero with a tiny speed, say 0.1 mph. Keep your speed in mph equal to the distance in miles that you have traveled from zero. How long will it take to cover 75 miles?

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#8

### Re: Average speed?

11/07/2006 3:31 AM

My "Keep your speed in mph equal to the distance in miles that you have traveled from zero" should have read:

"Once you have traveled 0.1 miles, start keeping your speed in mph equal to the distance in miles that you have traveled from zero".

Otherwise, it's a bit non-sensical!

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#13

### Re: Average speed?

11/07/2006 10:32 AM

Then the problem becomes;

dx = xdt

x = v = Aet; initial conditions;x = 0.1et

for x=75, t=ln(750)=6.62 h

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#20

### Re: Average speed?

11/08/2006 1:11 AM

Cameo, when you wrote: "for x=75, t=ln(750)=6.62 h", you were basically right, except that I think you forgot the first 0.1 km that took 1 hour!

So the answer should be 7.62 h, or am I missing something?

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#21

### Re: Average speed?

11/08/2006 6:05 AM

I assumed the vehicle is at 0.1 miles at t=0, that was what I understood.

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#22

### Re: Average speed?

11/08/2006 6:32 AM

Hi Cameo, I agree that your "I assumed the vehicle is at 0.1 miles at t=0, that was what I understood" is reasonable. I tried to state it slightly differently, but not quite successfully, it appears.

Incidentally, I did exactly the same thing in my eBook on Relativity and Cosmology, when dealing with the inflation epoch of universal expansion. There I started the size of the observable universe at one Planck length at the Planck time, with an exponential expansion curve. The cosmic inflation curve looks very much like the time/distance curve for the car. If you are interested, there's a free pdf download from this web page.

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#24

### Re: Average speed?

11/08/2006 12:13 PM

I have always found finding mathematical models for physical systems exciting. The system dx/dt = x is a first order, linear differential equation, but if this were a physical system, it would be an unstable and divergent one, because of the right-hand side pole located at 1.

So, you mean the universe is expanding to infinity as time goes to infinity?

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#25

### Re: Average speed?

11/08/2006 1:08 PM

Quote: "So, you mean the universe is expanding to infinity as time goes to infinity?"

Yep, that is approximately what present theory and observation predict!

But to explain it is a lo---ng story! If you want to dig into it, read some of the Cosmology stuff at the website I referenced before. (One do not need too much Relativity to understand Cosmology!)

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#26

### Re: Average speed?

11/08/2006 1:20 PM

I went to your webpage. The PDF link worked but you also have a link on that page to Wikipedia that is broken, just wanted to let you know.

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#27

### Re: Average speed?

11/08/2006 11:09 PM

Tx Roger, fixed it - seems I had the http// in twice!

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#28

### Re: Average speed?

11/12/2006 4:36 AM

Guru here is one more variant.

Start with the speed that is equal to distance you have alraedy covered.and accelarate @ miles remaining.

What will be your intial and final speed?

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#29

### Re: Average speed?

11/12/2006 9:24 AM

rakesh_semwal, your question is not clear. You wrote "Start with the speed that is equal to distance you have already covered. accelarate @ miles remaining.

What will be your initial and final speed?"

You have to give, what is called "initial conditions". For instance, if you start at distance zero and velocity zero, you will stay where you are, because acceleration is zero.

Further, if we use miles and mph as units, and you give a small speed as an initial condition, the acceleration ( in m/h2) will be very, very small indeed and it will take an almost infinite time to reach 75 mph.

So you must specify better - then maybe "cameo", who likes this type of problem, will give you a formula!

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#9

### Re: Average speed?

11/07/2006 8:08 AM

Speed is cycles per unit of time ~ RPM

Velocity is distance per unit of time ~ MPH

Your driving habits may limit your trip because someone will shoot you long before you reach your destination.

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#10

### Re: Average speed?

11/07/2006 8:56 AM

Speed is Velocity without direction.

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#11

### Re: Average speed?

11/07/2006 9:15 AM

I get 38 mph.... the sum of all the speeds, divided by their number = average. I'm curious!

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#12

### Re: Average speed?

11/07/2006 9:25 AM

That was the mistake I made. If you traveled at each of the 75 speeds for the same amount of time, rather than the same amount of distance, this would be the correct answer.

Look at it this way. For the 75th mile, you are traveling at 1 mph. So that takes 1 hour. The 74th mile, you are traveling at 2 mph, so that tales 1/2 hour. The 73rd mile takes 1/3 hour, and so on. So you get the series 1 + 1/2 + 1/3 +1/4 +...+ 1/75.

I've never calculated it, but the 4.9 hour answer is right, I believe.

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#14

### Re: Average speed?

11/07/2006 10:34 AM

okay agreed,

294.08 minutes 4.901355631 hours

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#15

### Re: Average speed?

11/07/2006 10:51 AM

294.08 minutes ≠ 4.901355631 hours, but I'm sure it is close :)

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#16

### Re: Average speed?

11/07/2006 11:22 AM

OK, the title isn't quite the same as "the question," but I did have a spreadsheet open at the time, so I had to check this out just to be sure...

A) for those who didn't read or understand the significance of the incremental speed restriction, we are NOT reducing our speed continuously to match the number of miles yet to travel...we are travelling exactly one mile at each speed. This does direct that the first entire mile is at 75mph, and the final mile is driven entirely at exactly 1 mph. Sure enough, the time driven at each speed, or the time spent driving during each mile, sum to Ttot=4.90135563055305 hours (or approximately 4h:54m:05s)

B) Since the incremental speed contribution of each segment is equal to 1 (75*1/75...1*1/1)
The forum title question can be answered by:

Savg ~ 75/Ttot = 75m/4.90135563055305h = 15.3 mph

...right? ;)

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#17

### Re: Average speed?

11/07/2006 11:30 AM

At the risk of starting an argument over significant digits, the answer can only be 4.90135563055305h, if you are traveling at exactly 75.0000000000000 mph for exactly 1.00000000000000 miles, and so on. 4.9 is close enough.

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#18

### Re: Average speed?

11/07/2006 4:33 PM

So...

Has anyone attempted to experimentally prove this?

I have often done the first leg at 75mph. The rest looks tedious. Though at rush hour I have appraoched the result.

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#19

### Re: Average speed?

11/07/2006 7:49 PM

Hmmmm...it really wasn't much of a "riddle" at all then. If I knew somebody that presented riddles such as this constantly, I would be less than amused...somewhere around the agitated/annoyed/homicidal band of things. Then again, I could just point out to the person that riddles are amusing due to the lateral thinking required to solve them, and that what he is presenting me with are basic exercises in mathematics. Excellent...

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#23

### Re: Average speed?

11/08/2006 10:02 AM

You'd kill someone over an intellectual puzzle? You must be great fun at parties.