Matters Mathematical

[neptune]

Gilbert and Sullivan wrote
I`m a mine of information on all matters mathematical.
I understand equations, both the simple and quadratical.
I`ve information vegetable , animal, and mineral.
I am the very model of a modern major general.

Well, I am looking for someone with similar abilities. My question is, is there a way to compare the retarding effect of a headwind , on a flat road, with the retarding effect of a gradient. So could we say for example that a 10MPH headwind is equivalent to a one in thirty gradient, or whatever gradient it turns out to be?

 

[GaRRy]

Well im sure its possible for a fixed set of figures But pretty certain its a moving target and certainly not a simple formula.

For starters these will effect out come
Weight ( more weight harder up hill but less effected by wind)
Drag coefficient ( more stream lined less effected by wind)
Air temprature ( cold wind denser so produces more drag)
Bound to be others.

 

[neptune]

You make some valid points Garry. Ok, we need to make a few assumptions. Lets assume a 10 stone rider, on a hybrid bike, so no dropped bars .And its the middle of summer, so lets assume a temperature of 13 degrees!I seem to think we have a member called 10mph who is a retired Physics Professor. This should be easy for him.

 

[flecc]

And for me! No, it's not possible to have direct comparisons for one simple reason.

Power required for gradients is linear with the gradient change and weight change.

Wind resistance is exponential, increases have to be cubed, i.e. double the speed, four times the resistance.
.

 

[jazper53]

Gilbert and Sullivan wrote
I`m a mine of information on all matters mathematical.
I understand equations, both the simple and quadratical.
I`ve information vegetable , animal, and mineral.
I am the very model of a modern major general.

Well, I am looking for someone with similar abilities. My question is, is there a way to compare the retarding effect of a headwind , on a flat road, with the retarding effect of a gradient. So could we say for example that a 10MPH headwind is equivalent to a one in thirty gradient, or whatever gradient it turns out to be?

Not being a Mathematician, but quite a logical thinker, maybe approach the problem by peddling into a 10mph wind, and measure the pressure required to maintain 10mph, once u have the data, find a hill with a known gradient, and measure the pressure required to reach 10mph, and with those two readings you have the start of your mathematical formula. good luck

Ps Maybe you could us a pulse meter to measure the required effort(s), should be able create a crude ratio scale with data, to start with.

 

[jhruk]

If you want to see the difference different parameters can make you can use one of the many online power calculators. Just search for bicycle power calculator.

Enter your data and they will do all the maths for you.

 

[flecc]

Simple measurment of that sort won't give a result that can be scaled Jazper, as I pointed out above, the climb factors of weight and/or gradient are linear, wind resistance is exponential.

In addition, its generally considered that up to 12 mph air resistance is negligible when cycling, and that takes care of most of the legal power assist range of e-bikes. It's only as we approach 20 mph that the air resistance becomes appreciable, but since all e-bike hill climbing is well below that speed, no usable comparisons are possible so the exercise would be pointless even if possible.
.

 

[neptune]

@flecc . I understand what you are saying . The only way around this is to assume that the cyclist maintains a given road speed . Let us assume that the speed is 8 MPH, because after all we are talking about headwind or up hill. So we might be able to say, if all the pre mentioned conditions are met , a 10 mph headwind is equal to a gradient of one in thirty, or whatever. So we could draw a graph of headwind speed versus gradient . If the road speed changed, or the riders weight, or the type of bike we would need a seperate graph, or at least a new curve on the graph .
@Jasper. I understand what you mean, but I don`t have any suitable measuring equipment. Here in the fens we have no hills, but I thought it would be nice if I knew the windspeed, to be able to say to myself, "today I am climbing a one in fifteen or whatever .

 

[flecc]

The trouble with measurement is that it's impossible to arrange consistent headwinds to measure, so any measurement must be in still air using a road speed to simulate a headwind. But as I've just posted above you, our speeds don't allow enough headwind to be simulated.

Hill climbing calculations are accurate and reliable for everyone, but air resistance ones aren't, simply because of the huge variables. A fairly short slim lycra clad rider using drop handlebars and a tall fat rider in loose clothing with sit-up-and-beg handlebars will have wildly differing air resistance factors.

Therefore I see no point in having a combined calculator for two such out of synch factors where the reliability of one is so let down by the inconsistency of the other. As jhruk has observed, there are online calculators for both separately and I think that is how they are best treated.

 

[10mph]

My question is, is there a way to compare the retarding effect of a headwind , on a flat road, with the retarding effect of a gradient. So could we say for example that a 10MPH headwind is equivalent to a one in thirty gradient, or whatever gradient it turns out to be?

I find the power calculator at Bicycle Speed (Velocity) And Power Calculator convenient for these sorts of calculations

Inputting your data with further assumptions:
rider 140 lb
bike 48 lbs
rider height 5' 8"
MTB
Wide Touring tyres
*temp 55F
*Height 10 foot above sea level
*pedal cadence 70/min
(* these make little difference)

I get the following at my 10mph cruising speed:

No head wind flat = 51 watts
10mph headwind flat = 142 watts
No head wind 2.4% slope = 142 watts

So at that a bike speed of 10 mph, a 10mph headwind on the flat is equivalent to a definitely noticeable 2.4% slope.

As Flecc points out the drag goes up non-linearily with speed, and at the ebike legal maximum of 15.5 mph I get:
No head wind flat = 147 watts
10mph headwind flat = 339 watts
No head wind 7.5% slope = 339 watts

So for a bike at the legal maximum of 15.5 mph, a 10mph headwind on the flat is equivalent to a challenging 7.5% slope.

In still air, the power you expend overcoming air resistance goes up as the cube of your speed.

When your speed is small compared with that of the wind the power to overcome the wind goes up as the square of the wind speed.

But these statements simplifies things too much, the calculators add together estimates of all the sources of resistance.

 

[10mph]

So we could draw a graph of headwind speed versus gradient . If the road speed changed, or the riders weight, or the type of bike we would need a seperate graph, or at least a new curve on the graph .

If you need help with making your graph let me know.

 

[neptune]

OK, so my initial theory was not really practical, but it gave us all something to think about on a wet Monday.70% of my road miles are on my pedal bike, rather than my ebike. I probably pedal50% of the time on my ebike. So the question was really more about pedal bikes. The fens are famous for their winds, which is why farmers` favourite cash crop these days is wind turbines. It is interesting to note that from `10mph`s post , that on the flat, at 10mph road speed, a 10mph headwind very nearly TRIPLES the required input power. Strange how we are less willing to waste energy when it comes from our own muscles. Todays daily ride into a 13mph wind with driving rain was a bit of a challenge, but when I turned round after 5 miles to come home, it got a lot easier .

 

[flecc]

Yes, you suffer in the same way they do in the Dutch poulders, strong unimpeded headwinds over large flat areas. In my very hilly and twisting high hedged lanes area, consistent headwinds rarely arise, it's more often sudden variable gusts, and it's the steeper hills or protracted gradual climbs that mostly affect my cycling.

 

[jazper53]

OK, so my initial theory was not really practical, but it gave us all something to think about on a wet Monday.70% of my road miles are on my pedal bike, rather than my ebike. I probably pedal50% of the time on my ebike. So the question was really more about pedal bikes. The fens are famous for their winds, which is why farmers` favourite cash crop these days is wind turbines. It is interesting to note that from `10mph`s post , that on the flat, at 10mph road speed, a 10mph headwind very nearly TRIPLES the required input power. Strange how we are less willing to waste energy when it comes from our own muscles. Todays daily ride into a 13mph wind with driving rain was a bit of a challenge, but when I turned round after 5 miles to come home, it got a lot easier .

Then maybe a heart rate monitor watch, could give measure of effort against headwinds, and if you ever find a hill in your part of the world maybe u can measure that as well

 

[flecc]

The fens are famous for their winds.

Thinking about this Neptune brings a different view on circumstances. Hills are always thought to be the main reason for having an e-bike and where the added power is most useful, but it's not unconditionally true. I think your circumstance is worse than mine.

When you go on a ride against a headwind and stay at your destination for any time, there's no guarantee of a compensating tailwind on your return. When I e-bike the seven miles to the high point of the North Downs, gradually climbing much of the way, I average about 11 mph. When I return I can guarantee to do it mostly at about 20+ mph with short bursts up to 34 mph.

So not only have you no guarantee of a compensating return, even when you do have one it doesn't realise the same overall gain that I enjoy.

 

[neptune]

Flecc, you made some very relevant observations there.I can only think of one occasion in the last two years of daily pedalling that I got caught out with a headwind both ways. And although it is not every day, I do get some home runs at 20+ mph, but no 34 mph bursts. Being retired and biking for recreation, I am in the fortunate position of choosing my destination each day, unlike a commuter. We have had an awful lot of easterly and north easterly winds this spring, but throughout the year, our most prevalent winds are westerly. This is fortunate, because on my westwind route I can cheat a bit . Outbound, I take the old road, which winds about a bit and is tree lined, mitigating the headwind effect . Coming home I ride on the hard shoulder of the bypass, a trunk road, not dual carriageway . Being a road that follows an old railway route it is straight as an arrow, and very exposed. So I get full benefit of the tailwind . The hard shoulder is as wide as a motorway one, and is commonly used by cyclists . Best of both worlds.

 

[flecc]

I do get some home runs at 20+ mph, but no 34 mph bursts.

But still pedalling of course. On my downhill return I don't even have to do that some of the time, I just let the gradient do the whole job, even for high speeds like the 34 mph, the big advantage of downhills hills over tailwinds.

 

[gerryscott]

just enjoy our bikes

 

[Davanti]

Coming home I ride on the hard shoulder of the bypass, a trunk road, not dual carriageway . Being a road that follows an old railway route it is straight as an arrow, and very exposed. So I get full benefit of the tailwind . The hard shoulder is as wide as a motorway one, and is commonly used by cyclists . Best of both worlds.

Provided you don't end up at the Pilgrim! (Or impaled on the Stump!)

 

[10mph]

Neptune:

Is this graph helpful? I have drawn 4 curves for bike speeds of 6 mph to 12 mph. 12 mph into a 30 mph head wind needs 633 watts - so that would not achieved by those on legal pedelecs.

The hill slopes are percentages, for example 10% means 50 foot climbed in a horizontal distance of 500 feet.

I hope this graph will be useful to the people in Lincolnshire in order to compare their speed on the flat into ferocious head winds to equivalent hill climb slopes about which which we in more hilly areas are inclined to boast.