Can anyone help resolve a physics argument (and prove me right)

[BillMasen]

I’m having a disagreement with my partner

Say there are 2 paths up a hill. One is steep but short, the other long but shallow. Both start at the same point at the bottom and end at the same point at the top

They say that the total energy expended taking the shallow path is less than taking the steeper path. I say it’s the same.

Who is right?

Shallow path easier. Steep will get heart pumping faster

Sorry but I'd agree with him

It's absolutely not the same.

I think the shallow path would be easier work, so less energy needed.

I think total energy should be roughly the same.

Are you married to my DH, OP? We've been having a very similar arguement discussion, but in our house it's frozen peas.
If you put 100g frozen peas in a pan and add boiling water to cook them, is it more energy efficient to add a small quantity of boiling water (will be cooled down more initially, but less to heat up again), or a large quantity (will retain its heat better, but may take more energy to get it boiling again).

I’d say that theoretically it’s the same. You have both gone from having 0 potential energy at the bottom of the hill, to X potential energy at the top. In practice though, the person taking the long, shallow route has probably had to overcome more friction, by simply walking further, but then the person on the shorter route will have had more friction per step due to the steeper angle…

I think the shallow path would use less energy.

I’m a hiker .

I think the total energy would be the same. I have had the same argument about running 5 miles vs walking 5 miles, without there being a definitive answer.

In terms of gravitational potential energy gained, they are the same.

However, it then becomes a question of which is the more efficient process. I would argue that you would use more input energy (have to eat more) to go up the long shallow path, as you have to lift your foot up each step, and there are far more steps.

The steeper path, however, would require more power (energy per unit time) as you would climb it more quickly, so would feel harder.

You are right OP. According to this website anyway!

You could go very slowly up the steeper path.

You can calculate it quite easily.

Here's some examples (just change the gradient to make it steeper).

isaacphysics.org/questions/walking_up_hill?stage=all

The change in gravitational potential energy is the same in both cases (assuming the same starting and end point), E = mgh where m = mass, g = acceleration due to gravity and h = change in height.

The work done is work = force displacement cos(angle) … the work done is the same even though the force needed to push the same mass up a steeper incline is higher.

www.physicsclassroom.com/mmedia/energy/au.cfm

DH says, theoretically it's the same due to potential energy. However - the shallower path will lose more to friction and the steeper path will lose more heat. So you can't tell.

@RoarySaury,

See my post above, that is the output energy, which is the same.

OP is talking about the input energy, and thus the efficiency of both processes. That is far less easy to estimate.

But running is definitely less efficient than walking, so running 5k burns a lot more calories than walking 5k.

I am also fairly certain that you would burn more calories on the long shallow path.

The peas question - it must be more efficient with less water, surely?

I often go on a walk/hike with my dogs where there is a very steep embankment.

I walked up it once from bottom to top in a perpendicular line and found it very strenuous and I was out of breath at the top.

If I walk It by ascending it sideways gradually getting higher every couple of steps it’s easy.

The process of walking is not about friction.

Each step, you raise your centre of gravity a little (working against gravitational force) and, simultaneously, move it forwards. You then topple over until you regain stability by placing your weight on your front foot and swinging your other foot forward.

Walking is really quite complex and is sometimes called ‘controlled falling’. If you watch a toddler you see the immense focus required to master it.

But it is the repeated raising of your centre of gravity which means you are doing work just to move forward on the flat.

Theoretically it's the same, but that doesn't account for the workings of the human body. You're both right in a way!

*I think the shallow path would use less energy.

I’m a hiker grin.*

me too and i agree

Shallow path requires less effort to lift them legs and move the body weight.

And peas are best put into hot kettle water and a lid left on top for three minutes without any heat underneath - the residual heat from the water will do the job for the least energy.

Thanks all

My argument is that it’s physics, and the energy expanded to lift a body the same height is (broadly) the same. Steep gradient uses more x shorter distance, shallower is less x longer distance.

I’m obviously ignoring friction, or the body’s efficiency, or anything else that may point to me being slightly inaccurate.

@Imissmoominmama

I think the shallow path would use less energy.

I’m a hiker .

Per step yes but you take more steps

@WaltzingToWalsingham

Are you married to my DH, OP? We've been having a very similar arguement discussion, but in our house it's frozen peas. If you put 100g frozen peas in a pan and add boiling water to cook them, is it more energy efficient to add a small quantity of boiling water (will be cooled down more initially, but less to heat up again), or a large quantity (will retain its heat better, but may take more energy to get it boiling again).

After getting fucked off at dh making veg that was still too crunchy I set out on a mission to test this out. About 1cm of boiling water in with the frozen veg, give it a wee turn over midway through, then fill the rest with boiling water. Books MUCH quicker.