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Same distance to the top of the hill

I think the reason the last caller on the 6/17 show had the same distance to the stop of the hill via the steep route or the circuitous route is that there is a level section of the route before she gets to the steep hill. Any other ideas?

Exactly - it is surprising that a couple of well educated guys (our heros?), trained in statistical principles, etc. would have failed to ask about the variances in grade between the 2 routes, given that the total distances are the same. Living in the mountains of NC apparently produces more common sense than MIT.

I think Mara’s Magical Mountain may not mean that she’s crazy just that she may leave out important information. What if the “steep” 3.1 mile route actually went up and over the hill and partially down the other side. She said both routes ended at the same place… she didn’t say that place was at the top of the hill.

OK guys. I get that you passed physics 101. But you are talking about an ideal sitation. Yes the work is the same if the efficiency is the same for all gradations. However, we live in a non-ideal world with car mechanics and car efficiency definitly up there with non-ideal world things. So, if the car’s efficiency is the same for all grades, you guys are correct. But if the car’s efficiency is not the same for all road grades, you all need to regroup.
Anyway, what do you expect from a bunch of MIT geeks?
Vicky Cornell geek 1980

Comments of Carparts and Seewa001 have merit, but I believe it is even simpler: she discovered the first wormhole on the planet. Or she’s crazy…

You forgot one imporant question. The caller said it was the same distance from her house to the top of the hill regardless of which road was taken. Assuming that the 2 routes to the top do not start at the same point, if the mileage is the same it is likely that she has to drive further to reach the straight up ascent.

It is like having a choice of two roads.  Both the same total distance Both starting from the same place and both ending at the same place and each road totaled 10 miles.  One might drive up a steep hill for 5 miles and down a equally steep hill for 5 miles.  

While the total amount of work is the same for both trips, the flat one might be able to maintain an even speed at an ideal gearing for the road, while the other trip would result in an up hill section and a equal down hill section. Chances are the best gear for that trip would not be the same.

One trip could mean a five mile level road with the engine working at it's most efficient, while  the other trip could force the use an a less desirable gear and reduce mileage.  

I would generally expect the trip over the level road will consume less total fuel than the one operating at less than ideal load would burn more fuel.

Option 1 - She’s crazy: No further analysis needed.
Option 2 - She’s dingy, inarticulate and was poorly questioned.
2a - NCMtnGirl: route 1 has a significant distance of level road before reaching the steep incline while route 2 gradually inclines over the entire route.
2b - Carparts: the destination is not the top of the hill. Route 1 inclines steeply and then descends to some point below the top while route 2 gradually ascends around the hill to the same point.

If either corollary of Option 2 is true, as stated already the work being done is the same but doesn’t necessarily result in the same efficiency in the vehicle. My guess is that the vehicle would operate more efficiently on the gradual grade and it would certainly be less stress on the engine (higher gear ratio / lower engine rpm).

So, again if option 2 is correct, I would have to side with her against her husband. blind hog; acorn…etc etc.

So we have 2 routes of same length to the same end point. The info did not say but I would suspect that the “steep” route was a “curved” route (not a straight line route) which went up a “side” of the hill and the other "route was more “uneven” direction wise. Critter

I have an explanation which noone has thought of so far,
so lets start with the facts:
the husband takes the steep hill up
the wife takes the circuitous route up
I completely disagree that both are exactly 3.1 miles.
the husband’s more direct route is shorter
the wife’s is longer
the husband’s car might have smaller-than-stock diameter wheels thefore his car “adds” or “counts more” (say it was supposed to be 16", he is driving on 15"
the wife’s car might have larger-than-stock wheels so the odometer counts less.
does that make sense?
They should drive the same car up and down both ways and then the other car up and down both ways and that would solve the marital disagreement!

what do you think?

If the distance is the same…from start to the end…then the “WORK” performed by each vehicle is the same.

The steeper grade road could NOT have been steeper the whole time…sorry but that’s physically IMPOSSIBLE IF the distance is the same. If it was steeper then the distance traveled would have been less. There may be a point where the grade was steeper, but NOT the whole road…it couldn’t have been.

If the less-of a grade road say had a steady grade of 2 degrees all the way to the top…and the steep grade road had (part of the road at 5 degree grade)…then at SOME point of the road it MUST have had LESS then a 2 degree grade…If it didn’t then the distance would have been shorter, which they say it wasn’t.

“I think the reason the last caller on the 6/17 show had the same distance to the stop of the hill via the steep route or the circuitous route is that there is a level section of the route before she gets to the steep hill. Any other ideas?”

My theory is that sometimes while driving uphill, you don’t realize it, especially if you drive a powerful vehicle. This can especially happen when driving through the mountains, and the grain of the rock face next to the highway doesn’t flow horizontally. The flow of the grain in the rock face can make it look like you are on flat ground or going downhill, when you are really going uphill, especially if you don’t notice the vehicle working harder.

There is a difference between Work and Energy. The net amount of Work done is force times distance. Assuming the distance is the same, then the Work done is the same assuming the speed is constant. That’s very different from the Energy used though. The Energy required depends on the efficiency the engine generates force at what levels. That would depend on the car I’d think.

There was a show awhile back where a lady discussed a similar problem where her husband said going up and down the mountain rather than around it didn’t make any difference since your potential energy was the same at the end. Tom and Ray got it right that time and said “Yes but you either have to use your brakes coming down off the mountain or you would be doing 80 mph at the destination.” I’m surprised they didn’t use the same argument here. They were the “Wackos” this time.

It would be nice if we knew where the top of this hill is, then we could google a map of the area and see if the roads are in fact the same as well as a lot of other details we are only guessing at right now.

Ok, I googled Poland, NY and I see a possibility. There is a road called Buck Hill Road. This may not be the road she is talking about, but it does present a solution. Buck Hill Road and two other roads form an almost perfect equilateral triangle. Lets say that the top of the hill is in the middle of Buck Hill Road. Now if she lives at the intersection of the other two roads, then the distance to the top of the hill would be the same.

It just so happens that one of these other roads is a highway. Lets assume that it is flat and level. When you turn off this highway at the south end of Buck Hill Road, you face a steep climb up the hill. Now if you take the back road, it may be a steady but shallow climb up to where it intersects with the north end of Buck Hill Road, so the climb from there to the top of the hill will also be shallow.

I didn’t look for a topographical map of the area so I don’t know if all this is true, but it could be. Of course, this would also mean that the caller was a bit misleading as her description made it sound like she left her house and went in a straight line to the top of the hill. No mention of other roads involved here.