# I have a question for the masses. if you were traveling at 60 mph

I have a question for the masses.
if you were traveling at 60 mph and were 60 miles from your destination you would be 1 hour away from your point.

when you reached 59 miles away from your point you drop your speed to 59 mph. you are 1 hour away.

when your 1 mile from your point you drop your speed to 1 mile per hour. you are 1 hour away
.... if you continue this for infinity you would always be 1 hour from your destination... i'm looking for some real help with this. is this correct?

phinds
Gold Member
2021 Award

No, this is NOT similar, really, to Zeno and the OP is correct that he would always be 1 hr from destination.

DaveC426913
Gold Member

...the OP is correct that he would always be 1 hr from destination.
No.

He can only claim to be "one hour away" from his destination under the proviso that he were to remain at the current speed for the rest of the journey.

He won't. It is not true that he is "an hour from his destination".

Yes, he is reducing his speed the closer he gets. He will never reach his destination. In fact, if we consider his change of rate of speed, we can confidently say "he is infinitely far (in duration) away from his destination" at all points.

phinds
Gold Member
2021 Award

No.

He can only claim to be "one hour away" from his destination under the proviso that he were to remain at the current speed for the rest of the journey.

He won't. It is not true that he is "an hour from his destination".

Yes, he is reducing his speed the closer he gets. He will never reach his destination. In fact, if we consider his change of rate of speed, we can confidently say "he is infinitely far (in duration) away from his destination" at all points.

DOH you're right, obviously

Thank you all for your posts. I'm sorry for my ignorance, I thought of this when i was 16 and have never asked the question to some "SMART" people before. Is there a name for this? or is it just a know point in the community that this is true? Thanks again.

DaveC426913
Gold Member

Thank you all for your posts. I'm sorry for my ignorance,..

Never apologize for not knowing something or for asking. A curious mind is a most treasured thing, especially around here.

Is there a name for this?

It is a convergent series.

An infinite series of numbers can add up to a finite number.

1/2 + 1/4 + 1/8 + 1/16 + 1/32 +1/64 + ... = 1

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russ_watters
Mentor

Actually, since he's changing his speed in 1mph steps, he will get there unless he changes his formulation...

He's approximating a hyperbolic approach, but only approximating.

Integral
Staff Emeritus
Gold Member

Sorry you will never arrive at your destination.

Figure out how far you go in each 1 min interval, sum them up. You will have traveled 30.5 miles when your speed reachs 0.

As the question is posed he does reach his destination, because the speed change only occurs at the mile markers. So at the last mile, 1 mile from home he slows to 1mile and hour. An hour later he arrives at his destination and changes his speed to 0mph.

As the question is posed he does reach his destination, because the speed change only occurs at the mile markers. So at the last mile, 1 mile from home he slows to 1mile and hour. An hour later he arrives at his destination and changes his speed to 0mph.
This sounds right to me. At each mile marker he's dropping his speed by 1 mph.

What that means is only that each mile takes progressively more time to cover, with the last mile taking a full hour.

Is there a quick way to add up the times and find out how long the trip takes doing it this way? I'm curious.

Working backward:

Mile 60 = 1 hr
Mile 59 = 1/2 hr
Mile 58 = 1/3 hr
Mile 57 = 1/4 hr
Mile 56 = 1/5 hr
Mile 55 = 1/6 hr
Mile 54 = 1/7 hr &etc

It's clear the bulk of the added time comes at the end of the trip and does not threaten to approach an infinite amount of time, or anything like that. Still, I'd expect the total time it takes to do it this way to be interesting in some way, shape, or form.

You are all wrong in your reasoning, a human is not some infinitely small point, he/she occupies a definite volume and space. Take it from the last mile, or 1760 yds, which is covered in 1 hour. Then going on decreasing the distance the time increases , 2 hrs for the half mile, 4 hours for the quarter mile and so on. Until finally when he reaches the last yard (0.8593 yds to be exact) he takes 2048 hours or 85.3 days to cross that small distance. The point being that he does make it, even if it takes a long time!

Is there a quick way to add up the times and find out how long the trip takes doing it this way? I'm curious.
I've got about 4.68 hours. Wolfram Alpha can do it in no time.

I've got about 4.68 hours. Wolfram Alpha can do it in no time.
Thanks, nasu!

Well, that figure isn't as interesting as I'd hoped it would be.

I have the answer for you. It is not a black or white answer and will surely piss off all those who like simple answers.

For the same reason that Schrödinger's cat is both dead and alive - a very real consideration in quantum physics, in your puzzle, the answer is that you will arrive at the destination eventually; however, it will take forever.

DaveC426913
Gold Member

You are all wrong in your reasoning, a human is not some infinitely small point, he/she occupies a definite volume and space. Take it from the last mile, or 1760 yds, which is covered in 1 hour. Then going on decreasing the distance the time increases , 2 hrs for the half mile, 4 hours for the quarter mile and so on. Until finally when he reaches the last yard (0.8593 yds to be exact) he takes 2048 hours or 85.3 days to cross that small distance. The point being that he does make it, even if it takes a long time!
Why did you stop at one yard? Why do you all along assume he's halving the distance and halving the speed, and then when it gets down to a yard, you declare he just magically leaps across that distance?

As for a human being a non-zero size, this is not relevant unless you get sloppy with your measurements. When he started his journey, his toe (or bumper) had not crossed the starting line, and he's not done until that toe (or bumper) crosses the finish line. This is true whether he is a point, a person, a car or a whale. The point of measurement is indeed a zero-dimensional point.

Talk about being "wrong in reasoning". You did it twice in one paragraph.

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sophiecentaur
Gold Member

It is up to the Engineers amongst us to say those immortal words "Near enough for Jazz". If you can't measure the separation distance then you've arrived.

DaveC426913
Gold Member

It is up to the Engineers amongst us to say those immortal words "Near enough for Jazz". If you can't measure the separation distance then you've arrived.

Agreed. If this were an Engineering Forum. But it ain't. We eat zero-dimensional points for breakfast.

sophiecentaur
Gold Member

Agreed. If this were an Engineering Forum. But it ain't. We eat zero-dimensional points for breakfast.

Indigestion for the rest of the day?

Eventually the object would come to a movement so slow it wouldn't even matter.

DaveC426913
Gold Member

...it wouldn't even matter.
A physicist will look at you like to just sprouted antennae.

Imagine a physicist saying...

"Well, we use a microscope to look at smaller and smaller things until we get down to these atom-thingies. That's small enough. Anything smaller doesn't really matter does it?"