 # 9.6.0 Ride Indexing Explained: Math

Symbols:

```          av = average speed of a ride measured while the wheels are turning.
(mph or kph)
C = 1.37E-3.  Calories (kcals) burned by the human body to raise one
pound mass one vertical foot.  This assumes a body efficiency of
24.9% and a transmission efficiency of 95%.
OR
C = 9.91E-3.  Calories (kcals) burned to raise one kilogram one
vertical meter.  This assumes a body efficiency of 24.9% and a
transmission efficiency of 95%.
d = distance traveled. (miles or kilometers)
D = the Calorie-equivalent amount of vertical climbing in feet or
meters to one mile or kilometer of flat-land riding for a
particular cyclist at a particular speed (mph or kph).  D is
called the cyclist's divisor.
F = 1 assuming no energy is expended while coasting down a 6% grade.
However, since the body expends some energy, F is set less than
one. ([unitless])
g = gross climbing. (feet or meters)
index = Calorie-equivalent flat-land riding miles or kilometers at
speed v.
irp = Index Rate of Progress, the average index speed of a ride
including all stops and rests. (mph or kph)
K = Calories burned for one mile or kilometer of flat-land riding at
speed v.
mirp = while-Moving Index Rate of Progress, the average index speed of
a ride measured while the wheels are turning. (mph or kph)
n = net climbing (feet or meters), current elevation minus
starting elevation.
t = total time including stops for an entire ride, ending time minus
starting time. (hours)
TC = total Calories burned for whole ride.
v = normal instantaneous cruising speed on flat ground without
ambient wind and in the chosen riding position on the bicycle.
This should be a speed that can be comfortable maintained for an
extended period.  (mph or kph)
W = total weight of the cyclist. (pounds mass or kilograms)
```

C is given.
av, d, g, and v may be read directly from an Avocet 50 cyclcomputer. av is
the value stored in the "average speed" buffer, and v is the instantaneous
velocity when riding under the conditions described above.
D is calculated.
F is guessed.
index is calculated.
irp is calculated.
K is calculated.
mirp is calculated
n and t may be calculated from data provided by an Avocet 50 cyclecomputer at
the beginning and at the end of a ride.
TC is calculated.
W is measured.

Note: A dietary calorie is equivalent to a kcal or Calorie.

K can be calculated with the assistance of Ken Robert's bike_power program.
Enter arguments for the following options: -wm, -wc, -a, and -v. For -wm,
enter the weight of the bicycle and accessories (water bottles, clothing,
kickstand, etc.) For -wc, enter your unclothed body weight. For argument -v,
use v, your normal instantaneous cruising speed on flat ground, a speed that
can be comfortably maintained for an extended period. Do not include stops.
Finally, enter a quadratic coefficient of air resistance (-a) that reflects you
r
typical riding position:

```			  position          a
----------        -----
standing       0.36
hoods or top       0.27
on the drops       0.172
tuck       0.145
drafting       0.12
```

If SI units are used, the +M option must be applied first and the -M option
must be applied last. See examples.

Take the Cal/hr figure returned by bike_power in the right-hand column and
divide by the argument you supplied for -v. This is K, your calories burned
per mile or kilometer of flat, windless riding at the chosen speed, v.

D is calculated by the following formula:

[English]:

D ft/mi = (K Cal/mi) / ((C Cal/(ft*lb))*(W lb))

[SI]:

D m/km = (K Cal/km) / ((C Cal/(m*kg))*(W kg))

or

D = K / (C*W)

F is determined by educated guess. The purpose of F is to allow some credit fo
r
expending energy on the downhills. The more you pedal on the downhills, the
lower F should be. If you only coast down hills, then F = 1. If you pedal
lightly or sometimes, then a reasonable value might be 0.9. If you hammer
down the hills, you might use F = 0.7 or so. F is always less than one.

Total Calories burned accounting for climbing and net climbing on an entire
ride is determined by:

[English]:

```   TC Cal = (K Cal/mi) * [(d mi) + (n ft) / (D ft/mi)
+ (g ft - n ft) / (2*F*(D ft/mi))]
```

[SI]:

```   TC Cal = (K Cal/km) * [(d km) + (n m) / (D m/km)
+ (g m - n m) / (2*F*(D m/km))]
```

or

TC = K * [d + n/D + (g - n)/(2*F*D)]

The first term, d, gives credit for distance covered. The second term, n/D,
gives credit for hills climbed but not descended, the net climbing credit. The
third term, (g-n)/(2*F*D), gives credit for gross climbing and descending
according to descending style, F, and deducts net climbing credit awarded by th
e
second term.

Divide through by K to obtain the index or equivalent flat-land miles.

index = d + n/D + (g - n)/(2*F*D)

For loop rides, rides that start and end in the same place, n equals zero, and
the formula simplifies:

TC = K * [d + g/(2*F*D)]

index = d + g/(2*F*D)

In addition to index, other useful quantities are "Index Rate of Progress", irp
,
and "while-Moving Index Rate of Progress", mirp. These can be calculated with
the following formulas:

[English]:

irp mph = (index miles) / (t hours)

[SI]:

irp kph = (index km) / (t hours)

[English]:

mirp mph = (index miles) / [(d miles) / (av miles/hour)]

[SI]:

mirp kph = (index km) / [(d km) / (av km/hour)]

or

irp = index / t
mirp = index / (d / av)

irp and mirp can be used to check the accuracy of index. Here's how:

Ride two rides whose indices are equal but where one has much climbing and the
other has little or no climbing. Ride both rides at normal pace. If the
indices are accurate, irp from the first ride should be close to irp from the
second ride. Also mirp from the first ride should be close to mirp from the
second ride. If a higher irp or mirp has been observed for the ride with
climbing, v may be too low. Conversely, if the ride with climbing has yielded
a lower irp or mirp, v may be too high. By adjusting v up or down and then
recalculating D, a more accurate index formula can be found.

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