| **The Nos. on how much Gram-Geeking really helps (long)** | AllUpHill
*Jun 4, 2002 9:32 PM* | | Just in case you ever wondered ... I did a little number crunching to hopefully shed some light on the subject. I wanted to look at the difference in improvement that riders of different weights and power outputs would see. It's just some kindergarten level physics to give an idea of how things relate ... the energies are strictly the potential gained going up a climb. For simplicity I just ignored other factors (wind drag, rolling resistance...) which of course matter when you ride, but are outside the scope here.
Feel free to check my work or comment on anything. I'm sure plenty of others have worked out this kind of thing before, but I've never run across any of it (or bothered to look). For those who are not real fond of numbers, I'll try to keep things clear but ask if you'd like a little more explaination.
The underlying idea is t = M*g*h / P (M=total bike & rider mass, g = acceleration of gravity, h = elevation gain, P = average power output, t = time. Note the time comes out in seconds if you use the standard units).
For all of this we'll consider a climb that gains 3000ft (914.4m) in elevation. If this large climb were 10% the whole way (ouch) it'd be about 5.7 miles. Let's look at a 135lb(61.23kg) rider (me) with a 20lb bike and (70.31kg total) and then with a 15lb bike (68.04kg total). With the heavier bike this will cost him 630056J of energy and 609715J with the lighter. Assume he sustains 200W of power (just made that up...I have no idea what I'd put out). That's 52.505 minutes versus 50.810, a difference of 1.695 minutes.
What about a heavier rider, say 170lb (77.11kg) with the same power output? Not very realistic as you'd expect a heavier rider of the same fitness to have a higher output, but just for the sake of experiment. He'd exert 772269J with the 20lb bike and 751928J with the 15 pounder. Should take 62.661min versus 64.356 min, a difference of 1.695min also.
The derivative (with respect to total mass) of the formula is simple: dt/dM = g*h/P. It's the rate of time gain per unit of mass that you drop. It's interesting that it's not a function of the mass in question, and yet it's inversely related to the power output. Given a certain amount of weight savings from that $250 magnesium stem, folks with lower power output will gain more time on a climb, independent of how much they or their bike actually weigh. Presumeably lightweights will output less power than heavier riders of equal fitness (let's say equal power/weight ratio) and will benefit more from gram-shaving.
By the way, I keep talking about a 5 lb difference. What about the 100g you might save with a seatpost upgrade? dt = dM*g*h/P. For this very large imaginary climb, and putting out 200W, it should give you back an incredible **4.5 seconds.** Less if you're working harder, but then you'd be going faster to begin with. *If said 170lb rider on a 20lb bike spent more time on the trainer this winter instead, maybe he could average 205 W instead of 200, and he'd take only 62.8 minutes not 64.4. That's over ***90 seconds** better without buying a new gizmo.
Back to the heavier rider but with the same power/weight (body weight not total weight) as Shrimp Boy, whose ratio was 3.266W/kg. The 170 lb rider has to spend the same energy this time (772269J or 751928J) but now he's cranking out 251.85W. Should take 49.760min versus 51.106min. As expected, the time difference is less (1.346 min) because of the higher power output. The weird thing is that these times are less than the respective times of the smaller rider who had the same power/weight...shouldn't they come out the same? No, because we're talking about body weight as I said. For the heavier rider, the 20lb bike is a smaller fraction of non-power producing weight than it is for the lighter rider. If the two rider |
| **continued...** | AllUpHill
*Jun 4, 2002 9:43 PM* | | ... If the two riders had weightless bikes (running?) they would theortically come out the same. Again, it's more important for smaller riders than larger riders to pare down the bike. Also holds for any spare baggage on the waist -- it's weight that ain't making power.
This is still a rather strange result though, that riding the same 15lb bike, the 170lb rider takes 49.8 minutes while the 135lb rider with the same power/weight ratio (and we might think ability) takes 50.8 minutes. I think it's right that our intuition and experience says that lighter riders generally climb better. Lighter riders can more easily develop a higher power/weight. I suppose if you could put two riders of different weight and about the same initial fitness through a certain period of similar and equally tough training, the lighter rider's sustainable output would be less (so a greater benefit from titanium valve stem collars :-), but not by all that much; he'd have a considerably higher ratio. |
| **Didn't understand a word of that!** | hayaku
*Jun 5, 2002 2:53 AM* | | But I have an 11km TT, climbing about 900m on Sunday. I weigh 75kg and my bike is 7.6... Can I post my result and get you to work out my power output???
I'll put it this way geek... Do it or I'll beat ya up! ('.')b
an interesting read though, thanks.
M. |
| **I understood it, and it depresses me.** | McAndrus
*Jun 5, 2002 5:33 AM* | | It says I have no hope of every being able to climb with the local climbing studs. I weigh 155 lbs and one of the better locals probably weighs in at 150. On an eight-mile, eight percent climb we use, he beats me to the top by at least ten minutes. (It takes me ~50 minutes to make the climb.)
If I get new wheels and drop 200 grams and get a new stem and drop 100 grams and get a new saddle and drop another hundred grams I will have layed out, oh $1,000, and will gain backed from climber-stud less than 1 minute out of a ten minute gap.
Sigh .... I guess it's true that I can't buy legs. |
| **Didn't understand a word of that!** | AllUpHill
*Jun 5, 2002 9:55 AM* | | It would be easy to work something out from those numbers but it would only reflect your power output in working against gravity, know what I mean? In addition you'd certainly put out a good deal more against wind drag and such. Maybe the website Doug gave below could give a more complete calculation to take drag into account.
As an aside, the point of my post was not so much for figuring out exactly how much time you'd save, or how much power you're giving out (again that site looks like it could do a nice job of that), but mainly to give an understanding of who will be most affected by weight and power changes. The lighter or heavier riders? Those who have less or more power output? |
| **interesting, but** | DougSloan
*Jun 5, 2002 5:36 AM* | | http://www.analyticcycling.com has all the formulas built into nice little plug in calculators.
Bottom line: for a given power, lower mass is faster. Right? Also, the lower the power, the greater the effect of lower mass. |
| **more about power improvements** | AllUpHill
*Jun 5, 2002 9:37 AM* | | Originally I just touched on how much time improvement the rider would see for a tiny increase in power. To go into a little more detail, the derivative of the time formula w.r.t. power is fairly interesting: dt/dP = -M*g*h/P^{2}. This is the time improvement you get for each watt of power more you can put out.
It's a negative value as you'd expect: increase your power output, decrease your time on the climb. It's grows (in magnitude) with total mass, meaning that a heavier rider/bike will gain more time from a given power hike than a lighter rider.
It's also an inverse-square relation to the current power output, meaning that if you're putting out more power to begin with, you'll gain less time for each added watt than someone putting out less power. That kind of makes sense... less powerful (maybe weaker, maybe less experienced, maybe smaller built) riders will get faster *faster* given each added watt output. The higher you've already developed you output, the more output you'll have to gain in order to better your times.
I think these principles can partly explain why it's easy for beginners to get moderately strong (you know, enough to hang with the typical local riders) and difficult to get really strong (to hang onto your local cat 1 or pro big shot). If you're heavier and presumeably climbing not real fast, you're going to improve your time relatively rapidly for a given power improvement. If you're new and don't have a real great power output at present, your time gains for each added watt will be better than when, after a time, your power gets on up there. The rich don't get richer as easily, so to speak.
And all of this doesn't begin to touch on how your training time and effot investment translates into power improvements and such ... that would be a considerably complicated topic which I have no idea about since it's a biological/physiological consideration. |
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