*The author has based this section in part on
the French book, "Guide du Vélo en Montagne" published
by Altigraph, as well an Internet article by Rainer Pivit, originally
published in German in Radfahren magazine, and translated by Damon Rinard. *
* Please write if you spot errors of fact or emphasis, or have suggestions.*
As a cyclist, you have an intuitive understanding
of the conditions that affect your riding speed. The following mathematical
treatment and discussion, therefore, is given simply as an intellectual
exercise. Perhaps you will enjoy knowing the "whys" of riding
speed—and therefore distance. Perhaps you will understand better
some of the dynamics of road bike racing, and of long distance touring.
**Your biking speed depends upon your continuous power output to overcome
the forces that slow down your bike: friction, air resistance, and (when
going up hill), gravity.** In stop and go city riding, and in some racing, your power is
also used to accelerate up to speed, but for bicycle touring and long-distance racing,
this is a negligible factor, and we shall ignore it here.
This discussion uses mainly metric units, most of which should be familiar.
Examples are given in both metric units, in black print, and American
units, in red print.
Watts (like horsepower) are a measure of power. In the formulas below "Wrider" means the
number of continuous watts of power put out by the efforts of the cyclist.
1000 watts = 1.34
horsepower. "Power" measures "force" times
"distance" per unit of "time". Example:
Moving an object against a resistive force of 1 pound, for a distance
of 10 feet, in 1 second takes the same power as moving it against a force
of 10 pounds for a distance of 1 foot in one second, or against a force
of 100 pounds for 1/10 of a foot in a second. For a bicycle, it
is evident that, using different gear combinations, a cyclist can apply
the same power input to either go at high speeds (long distances/second)
against low counteracting forces, or at low speeds to overcome high counteracting
forces.
Overall Formula:
x means "multiplied by". ^2 means squared, i.e., (V+Vwind)
x (V+Vwind) = (V+Vwind)^2. The C's in the formula below are various constant amounts,
which vary according to the input units used. Details are discussed below.
Wrider = Cfriction x V x P + Cair x (V + Vwind)^2 x V+ Cslope x P x Slope%
x V + acceleration
[Note that V (velocity, i.e., riding speed, i.e., distance ridden per
unit of time) can be factored out of all the terms on the right side and
left side of the equation; the terms of the equation remaining after this
is done are the various forces that the biker must apply and overcome.]
Using the above equation, if we know the total power put out by a rider
(Wrider) the coefficient of friction of the bicycle (Cfriction) the weight
of the rider (P), the component of the Wind speed acting against (or in
favor of) the rider, and the percentage of slope (Slope%), (and if we
assume that the rider is not constantly braking and accelerating up to
speed) we can calculate the speed (V) that will be attained by the rider,
and therefore the distance he will cover in an elapse of time.
Now let us examine the power required on a more detailed basis.
Power of the Rider:
A completely inexperienced rider, for long periods of time, can output
50 or 100 watts of leg power; whereas a *Tour de France * racer is
said to be able to generate 500 watts or more of continuous power—still
not up to a horse, but mighty impressive, none the less!
Experience teaches cyclists how much power they can put out on a sustained
basis. Some riders may choose to use a heart rate (pulse) monitor as a
supplement to their experience: during the course of a ride pulse correlates
directly with power output (though not over weeks or years, as aerobic
capacity may change).
If there were no forces resisting the cyclist, even the inexperienced
rider could accelerate a bike indefinitely—up even to rocket speeds.
But there are resisting forces.
Friction:
At low speeds and on flat surfaces and with no wind, the only resistance
that counts comes from friction. That friction goes up proportionally
to speed and to total weight. If you have ridden at a gym a stationery
bicycle that uses a frictional brake, you know that the total effort required
goes up proportionally to speed.
It is less obvious that friction should go up linearly with total weight.
In fact, it doesn't exactly. The major components of friction, however,
do rise more or less linearly with weight, such as the friction of the
wheels on the hubs and the rolling resistance on the road. Compared to
these, the resistance caused by pedals, pedal bracket, chain and derailleur
are minor. As a first approximation, then, we put down the following formula:
Watts_to_overcome_friction = 0.1 x V x P
When the velocity V is measured in meters/second and the weight P is
measured in kilos, for a decent quality bike, with well-inflated racing
tires on a smooth road, the metric Cfriction is said to be about 0.1. *We can calculate, therefore, that at 5 meters per second, a rider and
bike weighing 80 kilos faces a frictional resistance of about 40 watts. *Poor quality bicycles, heavy loads, inefficient tires, and especially poor road surfaces can substantially
increase the coefficient of 0.1, whereas the highest quality bikes on
smooth roads may have coefficients as low as 0.08
One meter per second is 3.6 kilometers per hour
or 2.237 miles per hour, and one kilo is about 2.2 pounds. So in American
units, roughly:
Watts_to_overcome_friction = .02 x MPH x Pounds.
For our 176 pound rider plus bike going 11.2 miles
per hour we calculate 39+ watts of resistance, approximately the same
result as above.
Light weight riders are at an advantage over equally strong heavier ones. Similarly,
riders with lighter bicycles and lower touring loads have an advantage.
This advantage is proportionally more obvious at touring speeds(under 25 km/hr - 15 mph),
where air resistance is not much of a factor.
*If friction were the only resistance, a typical untrained rider could
zoom along at 28 miles per hour, putting out 100 watts to do so. The better
the bicycle and the better the road surface, the faster would be his or
her speed. But air resistance most definitely comes into play.*
Air and Wind Resistance:
Air is a "fluid", so to speak, though a thin one. When you
move through a fluid faster, it puts up much more resistance. If you have
been swimming, you know you can move your hand very easily in the water
if you do so very slowly; but try to move it faster, a huge force is required..
The same is true, as any experienced cyclist knows, in the air. At a
few miles per hour, (assuming no wind), you barely feel air resistance,
but at 15 miles per hour, it pushes strongly against you. The resistance
of fluids—certainly in the case of the wind—goes up with the
square of the velocity, and the faster one goes the more air resistance
one encounters. Thus a 10% increase in speed requires a 33% increase in
power, to overcome air resistance and a 25% increase in speed requires almost a doubling of power.
The formula for the power to overcome air resistance is:
W_to_overcome_air_resistance = Cair x (V + Vwind)^2 x V. If there is
no wind, it is simply: Cair x V^3.
For the metric system, Cair ranges from perhaps 0.8 for a city bike without
baggage or a hybrid with baggage, to .7 for a mountain bike or loaded
touring bike to perhaps 0.45 for a hybrid or upper position on a racing
bicycle without baggage, to as low as 0.36 for full racing position on
a conventional racing bike. (*The numbers for the racing bicycle and
the mountain bike, both without baggage, come from a study. Those for
the unloaded city bike and the unloaded hybrid are guesses.)*
Thus, a rider using a hybrid or the upright position on a racing bike,
traveling at 5 meters per second (11.2 mph),
will need to expend 56 watts to overcome air resistance. In racing position
on a racing bike, the rider need to expend only 45 watts. A rider using
a mountain bike may need to expend almost 90 watts to go the same speed.
In American units, Cair ranges from approximately
0.062 for a city bike or fully loaded touring bike to perhaps 0.04 for
a hybrid, and 0.032 for racing position on a racing bike.
At 11.2 miles per hour (5 meters per second) on
a hybrid bicycle, we calculate (as before) 56 Watts of resistance.
Therefore, at 11.2 miles per hour, approximately
slightly more than one-half of all power generated is spent overcoming
air resistance—more on a mountain bike, less in racing position on a racing
bicycle.
So, suppose you want to go 25% faster? You need to put out 95% more power! Well, at slower speeds, 12 mph, not really. Because half of your original
power was used to overcome the force of friction, and only the rest of your
power needs only to increase 95%. To go 25% faster (16 mph) you need
to increase your power by about 60%. So instead of a novice's 100 watts of power you need to put out 160 watts of power. To ride your hybrid at 20 mph you might need to put out 360 watts of power.
At 20 mph, four-fifths of your total power is already spent overcoming
air resistance. To go 25% faster, you need to increase your total power
by 83%.
The previous paragraphs assume no wind. If there is a wind, we need to
consider it, but not the total wind speed; rather only the portion of
that speed that is against or behind the rider. Obviously, if the wind
is directly behind a rider at the strong speed of 20 miles per hour, that
rider will be able to ride much faster. If that rider could output 90
or 100 Watts of power, and thus ride about 12 mph in no wind, he will
now be able to ride at perhaps 24 miles per hour. (At that speed he will
have twice the frictional resistance but only a bit of wind resistance.)
On the contrary, if the wind is totally against this rider at 20 miles
per hour, a biking speed of about 3 miles per hour for this occasional
biker brings the equation into balance.
If a Tour de France racer can continuously put out 5 times the power of a novice
biker, then he should be able to ride about twice as fast on the same
bike , that is, for (say) a hybrid bike, about 24 miles per hour rather
than 12 (about 40 km/hr rather than 20). The racer of course actually
goes faster than this: because his bicycle and clothing and position are
more aerodynamic; and because, also, he usually rides in a pack that greatly
lowers air resistance.
Gravity Climbing Slopes
While climbing steeper hills gravity becomes important, and air resistance
becomes unimportant.
It is easy to see why: On the way up slopes, gravity greatly reduces
speed, and at low speeds, air resistance is insignificant.
The formula for Gravity is: W_to_overcome_gravity = 9.81 x P x %slope
x V.
Where Cslope is a coefficient, P is your weight, and V is your speed
on the road.
The faster you go, the more you weigh, and the steeper the slope, the
more power is required to take you up the hill.
The "percent of slope" technically measures
the altitude gained per horizontal distance. Although this measure may
be the one you see in signs, documents and formulas, it is not very useful
in everyday bicycle touring. It measures the base of the triangle, not
the hypotenuse along which runs the road.
For every day road biking and for use in this formula,
we measure the altitude gained per distance of road. Maps or an altimeter
show you the elevation gain, and your trip computer shows you the distance
ridden. For low percent slopes, the two numbers are very, very close.
For higher grades, the two numbers are still quite close: An 18% grade
(extremely steep) measured from the road corresponds to an 18.3 % slope measured from the
base.
For a 6% slope, in metric units, the formula uses weight in kilos, speed in meters per
second, and a constant of 9.81. A rider weighing 80 kilos (with bike),
riding at 2 meters per second (7.2 kilometers/hour) must
generate 94 Watts of power to overcome the force of gravity.
In American units we use pounds, miles per hour
and a constant of 2. Thus the same rider and bike, weighing 176 pounds,
and traveling at 4.47 miles per hour expends 94 Watts of power to overcome
the force of gravity, as previously calculated.
At this speed (7.2 kilometers/hour - 4.47 miles/hour) our rider only
needs to expend 16 Watts to overcome friction. Assuming that there is
no wind, air resistance consumes less than 4 Watts. A total of 114 Watts
of power is needed to climb the hill at this speed. That is the most this
hypothetical novice rider can muster.
If the hill is twice as steep—12%, this rider can only proceed at
3.6 kilometers per hour, or about 2.25 miles per hour. He must use his
lowest mountain-bike gears. Maybe he would be better off pushing his bike
up the hill!
Most Alpine Cols (passes) in France have average slopes of between 6%
and 10%.
Descending Slopes
**On curvy mountain roads, total speed will be limited to what is safe.
Power and gravitational acceleration make no difference. **
**In straight line descents** without pedaling, on wide roads (ignoring the
effects of wind), bikes will accelerate until wind resistance plus frictional
resistance equals the acting force of gravity.
We take the general formula above, set the power input to zero, drop
out the terms for the wind, and put the coefficient for gravity as a negative
number, since gravity will be helping the rider, rather than opposing
him.
0 = Cfriction x V x P + Cair x v^2 x v - Cslope x P x% slope x V
Since V appears in every term on the right side of the equation, and
since 0 is on the left, V can be dropped out of the equation. The equation
now represents forces, rather than power.
In metric units, 0 = Cair x V^2 + 0.1 x P - 9.81 x P x %slope. Transposing
and factoring terms, we have:
V^2 = P / Cair x (9.81 x %slope - 0.1)
For the 80 kilo rider (including bike and baggage) on a 7% slope, with
a 0.45 aerodynamic coefficient (hybrid) V will be 10.21 meters/sec = 36.8
kilometers/hour = 22 miles per hour. On a 12% slope, the speed obtained
without pedaling and no wind would be 52 kilometers/hour or 31 miles/hour.
If the individual rider reduces air resistance by 20%, to a coefficient
of 0.36, he will descend slopes of 7% and 12% at speeds that will be respectively
11% and 8% higher; i.e., at 41 and 56 kilometers per hour (25 and 34 miles
per hour).
On the 12% descent with a hybrid or touring bicycle, even if a rider wished to add pedal power, at 35 mph his bicycle
is unlikely to be equipped with the gears to do so.
Tail winds and head winds can considerably effect the actual speed of
a descent.
Concluding remarks
Whereas, on the flat, and out of the pack, the bike racers with the most
power, the best aerodynamic position, and the most aerodynamic bicycle will do the best, even if the weight is
slightly more, when it comes to going up steep hills, the rider with the
best ratio of power to total weight will excel.
(Coming down straight, steep hills, the rider with more weight will gain
an advantage, but less so, as his speed does not pick up proportionally.)
For racers, it has always been obvious that it pays to
lose fat from body and bike, to improve aerodynamic efficiency, to cut
friction, and to increase personal power. Since most of the rest of us are the weight we are, and have the power we have, and
own the bicycle we own, the above mathematics are perhaps of theoretical
interest only.
We cyclists who do long-distance touring have learned not to
exhaust ourselves or strain our muscles by riding too rapidly against
the wind or up the steep hill. Experience can tell us better than any
calculations how much power and endurance we have available, and, over both short and
long intervals, what our optimum power output should be.
For further information on this subject see: http://en.wikipedia.org/wiki/Bicycle_performance
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