Advanced mathematics question - grab your Tylenol before reading

[mastery]

Buddy and I have some pretty in-depth conversations in which we discuss the most odd-ball things about the universe and try to figure out the answers or find solutions, but sometimes we get stumped. Example: https://www.indianagunowners.com/fo...-me-how-became-standard-protocol-tipping.html

Advanced mathematics question this time around that I figured somebody here would be able to help us with. Not sure how to word this, but I'll give it a try.

He knows a thru-hiker...that's somebody that walks long distance trails in their entirety like the Appalachian Trail for many months at a time carrying their lives in their ultra-light backpack. We got discussing gear carry weights and how the amount you carry actually becomes cumulative over distance; so by adding a few more things in the backpack, a person creates an exponential amount of energy required to carry those things over distance. How can we figure out the difference in total weight carried over a distance...in some form of measurable means? Example:

180 pound guy that carries a backpack and gear that weighs 20 pounds over 2,000 miles.

VERSUS

180 pound guy that carries a backpack and gear that weighs 25 pounds over 2,000 miles.

Simple thought says the difference is the second guy simply carries 5 pounds more. But mathematics says that over that 2,000 mile distance, he's actually carrying X amount more weight cumulative over the distance, which we figure must be thousands of more pounds in feeling totaled, and obviously requiring more energy to move that additional 5 pounds. Anybody that's humped a pack in the military knows that carrying a few additional pounds of gear becomes harder to carry later in the trek versus somebody else that has trimmed things out of their pack.

So it's not just a few pounds difference but the accumulated additional carrying weight over time and distance is what makes the those few extra pounds to be harder to carry. There's got to be a mathematical way to quantify both one person's carry bag over distance and time as well as the difference when comparing to a second person.

In the end, we'd like to see a visible difference. Making up numbers, it would show an answer like the first guy carried 200,000 pounds cumulative over the trip, where the second guy carrying five more pounds in his pack carried 300,000 pounds.

So the specific question:

What is the equation to show the difference in the example above?

And no, Mr. Foxworthy, I am not smarter than a fifth grader.

 

[eldirector]

Sounds like you are wanting to calculate the energy consumed carrying the two different weights over the same distance in the same amount of time?

A quick google gives us this:
https://www.brianmac.co.uk/energyexp.htm

There is a handy calculator about 1/2 way down the page. Its all metric units, but google can help you convert.

In your example:
A 180 lb guy with a 20 lb pack (200 lb total) = 91 kg
2000 miles = 3219 km
time to walk 3219 km at 4 kph (walking) = 804 hrs = 48285 minutes
=224525 calories

Add the heaver pack (now 93 kg):
= 229354 calories

About a 2% increase in energy consumed over that distance. Ignoring all of my rounding errors.

All of this ignores that biomechanics of carrying a backpack. Also ignores your body's ability to DELIVER that amount of energy in that amount of time.

 

[HoughMade]

...Also ignores your body's ability to DELIVER that amount of energy in that amount of time.

Which I think is the real issue here.

 

[spencer rifle]

Make it more complicated by figuring in all the ascents and descents, both of which take more energy than a level walk.

 

[eldirector]

IMHO - doesn't matter.

The delta will be the same if the only difference is weight.

Meaning, the exact same person walked the exact same distance, and the exact same terrain, in the exact same weather, carrying the exact same pack, with exactly 5 lbs more on the second trip (and the weight never varied as food/water consumed).

In any case: yes you can calculate it. No, you can't be accurate with all of the possible variables. But, if you control for those variables, you can get a baseline, and extrapolate from there.

 

[T.Lex]

180 pound guy that carries a backpack and gear that weighs 20 pounds over 2,000 miles.

VERSUS

180 pound guy that carries a backpack and gear that weighs 25 pounds over 2,000 miles.

To me, those numbers seem easy. The 5 lbs difference is 25% more than the lower amount. So, whatever his energy expenditure (totally ignoring the actual math upthread) it would be 25% more for the same distance/terrain. That means he would also run out of energy approximately 25% earlier (some estimation there, based on metabolism, baseline strength and endurance differences, etc.).

 

[Woobie]

Here's another equation: work used to move the pack.

Each pack is moved 10,560,000 feet.

Ignoring the body weight, one pack requires 264,000,000 ft-lb of work, while the other requires 211,200,000 ft-lb. the difference is 52,800,000 ft-lb of work.

 

[eldirector]

Here's another equation: work used to move the pack.

Each pack is moved 10,560,000 feet.

Ignoring the body weight, one pack requires 264,000,000 ft-lb of work, while the other requires 211,200,000 ft-lb. the difference is 52,800,000 ft-lb of work.

.
Moved vertically?

 

[Woobie]

.
Moved vertically?

Its just a linear equation, isn't it? I guess it doesn't make sense if you aren't using 25 lbs of force to move the pack. The pack is acting on your body with 25lbs of force the whole time though.

 

[mastery]

Make it more complicated by figuring in all the ascents and descents, both of which take more energy than a level walk.

That was discussed. Trying to factor in ground elevation changes would be near impossible, as many happen over a period of a quarter of a mile. However, onc could taken known mountains and hills heights and factor that in. Baby steps though, so we'll start with if we just factored that the elevation doesn't change at all.

 

[mastery]

About a 2% increase in energy consumed over that distance. Ignoring all of my rounding errors.

That makes sense. And a good way to try to put a measurement to the question.

 

[mastery]

To me, those numbers seem easy. The 5 lbs difference is 25% more than the lower amount. So, whatever his energy expenditure (totally ignoring the actual math upthread) it would be 25% more for the same distance/terrain. That means he would also run out of energy approximately 25% earlier (some estimation there, based on metabolism, baseline strength and endurance differences, etc.).

So, again we have a way to measure calorie expenditure for carrying the heavier load.

I'm trying to get my mind around how to convert the known calorie expenditure to what can be measured as how much additional weight "being felt" over the course of a hike may possibly be measured...if that's possible. And that part may not be able to measured, even if was the same person doing the exact same route, just with the different pack weights.

The calorie requirements of both people are now easily calculated with what's been mentioned already. So one "penalty" of having to carry extra load is the increase in calorie intake that is needed.

 

[JettaKnight]

There's a whole lot of non-mathematical issues - actually, the math is simple if you understand the physics.

There's the issue that others have brought up concerning the difference between energy and work, then there's the difference between mass and weight, then there's the difference between carrying a pack on your back while walking versus lifting the pack - the latter is an action that puts potential energy into the pack; the former is more complex an involves some kinesiology.

 

[mastery]

There's a whole lot of non-mathematical issues - actually, the math is simple if you understand the physics.

There's the issue that others have brought up concerning the difference between energy and work, then there's the difference between mass and weight, then there's the difference between carrying a pack on your back while walking versus lifting the pack - the latter is an action that puts potential energy into the pack; the former is more complex an involves some kinesiology.

We'll just keep is like two people of the same size, weight, and body shape are walking next to each other...each carrying the different weighted pack. Same terrain, same walking speed, same length of step, and both doing the same work of walking with their pack.

 

[eldirector]

Since I am sure this has been studied before, I hit up the Google machine again:
http://www.dtic.mil/dtic/tr/fulltext/u2/p010987.pdf

Little study of 0, 5 kg, and 10 kg loads in two different style backpacks. They were asking about military vs. load-bearing packs. They do have some interesting data on energy consumption at different pack weights, though.

 

[indytechnerd]

The physics of this aside, hiking wisdom says, as a general rule, 1000ft of elevation change = +1 mile equivalent. Looking at the physics, which requires more work, pulling a mass a certain distance along a level plane, or up a plane angled at 30º, when you want to maintain a minimum velocity (not stopping)? Going down that plane, gravity is still working against as you want to keep from exceeding a maximum velocity (falling on your face).

I would imagine the perceived effort for 25lbs vs. 20lbs would be enormous over time. I know I sure as hell would rather take 20lbs into the woods than 25.

 

[Woobie]

The physics of this aside, hiking wisdom says, as a general rule, 1000ft of elevation change = +1 mile equivalent. Looking at the physics, which requires more work, pulling a mass a certain distance along a level plane, or up a plane angled at 30º, when you want to maintain a minimum velocity (not stopping)? Going down that plane, gravity is still working against as you want to keep from exceeding a maximum velocity (falling on your face).

I would imagine the perceived effort for 25lbs vs. 20lbs would be enormous over time. I know I sure as hell would rather take 20lbs into the woods than 25.

I'm guessing 80 lbs is right out.

 

[spencer rifle]

Actually doing that on the AT is more difficult. We met a through hiker with the trail name Oz (as in the abbreviation of ounce). He carried a 20 lb pack, but without some IMHO essentials. Be borrowed medical supplies from us, as he had none of his own. He had no raingear. He mostly cooked on other hikers' stoves/fires. He had no personal care items to speak of, and we could tell. No tent, so he depended on shelters having room inside. Sure, a great concept. I would have liked a lighter pack. But to get down to 20, you have to have REAL expensive equipment, or leave things behind.

 

[indytechnerd]

I'm guessing 80 lbs is right out.

I have a guess as to where that 80lb pack originated, and while I admire the effort, I'm sure I can get most of those items at a considerable weight savings. Take food, for example. An MRE weighs in at about a pound and a half, I can get a day's worth of food in about half that weight.

 

[BehindBlueI's]

7.