Ride Indexing Explained by Bill Bushnell (bill.bushnell@pobox.com) and Chris Hull Copyright 1991 ------------------------ Introduction ------------------------ Bicyclists often wish to compare the relative difficulty of rides of varying lengths and of varying amounts of climbing. Today, one can buy inexpensive instruments that can accurately record gross climbing and distance traveled, two data values important for comparing different rides. The Avocet model 50 cyclecomputer is one such instrument. The aim of this article is to develop a system of comparing rides with different amounts of climbing and distance by determining a single parameter, an index. The index of a ride is the equivalent "flat-land" miles or kilometers in terms of Calories burned by the human body. A formula to determine the index of a ride will be constructed that is accurate enough to be useful yet does not require complicated calculations. That this formula can be modified to accommodate riders of different weights (masses) and of different riding speeds will be shown. An iterative means of checking and adjusting the formula will also be shown. Of course, the formula must be used consistently in either English or SI units. ------------------------ Assumptions ------------------------ 1) There is no wind throughout the entire ride. 2) Downhill grade is 6%. 3) Given accurate input, Ken Robert's computer program "bike_power" is assumed to produce accurate output. ------------------------ 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 comfortably 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 your 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 for 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 the 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. ---------------------------- Examples using English units ---------------------------- Finding the cyclist's divisor, D: First, calculate K. I ride with an average flat-land speed of v=17 mph. My combined bike and body weight is approximately 210 lbs., and I spend most of my time riding on the hoods or on top, so I use a quadratic coefficient of air resistance a=0.24. bike_power session: % bike_power -wm 42 -wc 168 -v 17 -a 0.24 grade of hill = 0.0% headwind = 0.0 mph weight: cyclist 168.0 + machine 42.0 = total 210.0 lb rolling friction coeff = 0.0060 BM rate = 1.40 W/kg air resistance coeff = (0.2400, 0) efficiency: transmission = 95.0% human = 24.9% mph F_lb P_a P_r P_g P_t P hp heat BM C Cal/hr 17.0 4.4 105 43 0 8 156 0.21 470 107 732 629 K = (629 Cal/hr) / (17 mi/hr) = 37 Cal/mi Now calculate D. D = (37 Cal/mi) / [(1.37E-3 Cal/(ft*lb))*(210 lbs)] = 129 ft/mi This means that for me, I use as much energy to lift me and my bike 129 vertical feet as I use to pedal one mile on flat, windless ground at 17 mph. I have decided to use F = 0.9 since I sometimes pedal on downhills, but not always. So my indexing formula is: index = d + n/129 + (g - n)/232 For the examples below, D will be accurate to +/- 1. In practice, given that indexing is an approximation, one can round the figures to the nearest 10 or even to the nearest 50 without too much loss of accuracy. One might use D = 150, and 2*F*D = 250 for in-the-head calculations. ---------------- Example 1: Suppose I take a 75 mile loop ride with 4000 feet of climbing. What is my index? For a loop ride, I use the following formula: index = d + g/232 The index for this ride is: index = 75 + 4000/232 = 92 This means I used the same amount of energy on this ride as I would use to ride 92 miles on flat ground at my normal riding pace. ---------------- Example 2: Suppose I rode my loop in 6 hours total time (including lunch and rests) and my Avocet 50 shows an average speed of 17.2 mph. Calculate the index rate of progress (irp), and the moving index rate of progress (mirp). irp = 92/6 = 15 mph mirp = 92 / (75 / 17.2) = 21 mph ---------------- Example 3: How many Calories have I burned on my loop ride? TC = index * K TC = 92 * 37 = 3404 That's almost a pound of fat! ---------------- Example 4: Suppose I start the ride at sea level and end at 1300 feet. What is my index? Since the ride starts and ends at a different elevation, we must use the complete formula. index = d + n/129 + (g - n)/232 index = 75 + 1300/129 + (4000-1300)/232 = 96 ---------------- Example 5: Suppose I've ridden 20 miles with 2500 feet of climbing in 2 hours. How long will it take me to ride the remaining 55 miles and 1500 feet of climbing at the same pace? (Assume that I am currently at my starting elevation.) My current index is 20 + 2500/232 = 31. My index average speed is (31 miles) / (2.0 hours) = 15.5 mph The index of the remaining ride is: 55 + 1500/232 = 61 miles. At my current index speed, I should be able to complete the ride in (61 miles) / (15.5 mph) = 3.9 hours or 3:56. ---------------- Example 6: I wish to compare the 100 mile and 200 kilometer routes for the 1992 Sequoia Century: Both rides start and end in the same place, so we can use the simplified formula. 100 mi route: d = 100, n = 0, g = 10,800 200 km route: d = 121, n = 0, g = 9,800 The index for the 100 mi route is: 100 + 10800/232 = 147 miles The index for the 200 km route is: 121 + 9800/232 = 163 miles Number of Calories burned: 100 mi route: K * 147 = 37 * 147 = 5440 200 km route: K * 163 = 37 * 163 = 6030 One might conclude that the 200 km route was more difficult. One should be aware that difficulty or pain is subjective, and the formulas given above take into account neither steepness of grade nor availability of efficient gearing for a particular grade. ---------------------------- Examples using SI units ---------------------------- Finding the cyclist's divisor, D: First, calculate K. I ride with an average flat-land speed of v=27 kph. My combined bike and body weight is approximately 95 kg, and I spend most of my time riding on the hoods or on top, so I use a quadratic coefficient of air resistance a=0.24. bike_power session: % bp +M -wm 19 -wc 76 -v 27 -a 0.24 -M grade of hill = 0.0% headwind = 0.0 mph weight: cyclist 167.5 + machine 41.9 = total 209.4 lb rolling friction coeff = 0.0060 BM rate = 1.40 W/kg air resistance coeff = (0.2400, 0) efficiency: transmission = 95.0% human = 24.9% mph F_lb P_a P_r P_g P_t P hp heat BM C Cal/hr 16.8 4.3 101 42 0 8 151 0.20 455 106 712 612 K = (612 Cal/hr) / (27 km/hr) = 23 Cal/km Now determine D. D = (23 Cal/km) / [(9.91E-3 Cal/(m*kg))*(95 kg)] = 24 m/km This means that for me, I use as much energy to lift me and my bike 24 vertical meters as I use to pedal one kilometer on flat, windless ground at 27 kph. I have decided to use F = 0.9 since I sometimes pedal on downhills, but not always. So my indexing formula is: index = d + n/24 + (g - n)/44 For the examples below, D will be accurate to +/- 1. In practice, given that indexing is an approximation, one can round the figures to the nearest 5 or even to the nearest 10 without too much loss of accuracy. One might use D = 25, and 2*F*D = 50 for quick, in-the-head calculations. ---------------- Example 1: Suppose I take a 100 kilometer loop ride with 2000 meters of climbing. What is my index? For a loop ride, I use the following formula: index = d + g/44 The index for this ride is: index = 100 + 2000/44 = 145 This means I used the same amount of energy on this ride as I would use to ride 145 km on flat ground at my normal riding pace. ---------------- Example 2: Suppose I rode the loop in 6 hours total time (including lunch and rests) and my Avocet 50 shows an average speed of 23.0 kph. Calculate the index rate of progress (irp), and the moving index rate of progress (mirp). irp = 145/6 = 24 kph mirp = 145 / (100 / 23.0) = 33 kph ---------------- Example 3: How many Calories have I burned on my loop ride? TC = index * K TC = 145 * 23 = 3340 ---------------- Example 4: Suppose I start the ride at sea level and end at 400 meters. What is my index? Since the ride starts and ends at a different elevation, I must use the complete formula. index = d + n/24 + (g - n)/44 index = 100 + 400/24 + (2000-400)/44 = 153 ---------------- Example 5: Suppose I've ridden 50 km with 500m of climbing in 2 hours. How long will it take me to ride the remaining 70km and 1500m of climbing at the same pace? (Assume that I am currently at my starting elevation.) My current index is 50 + 500/44 = 61. My index average speed is (61 km) / (2 hours) = 31 kph. The index of the remaining ride is: 70 + 1500/44 = 104 km. At my current index speed, I should be able to complete the ride in (104 km) / (31 kph) = 3.35 hours or 3:21. ---------------- Example 6: I wish to compare the 100 mile and 200 kilometer routes for the 1992 Sequoia Century: Both rides start and end in the same place, so we can use the simplified formula. 100 mi route: d = 161, n = 0, g = 3300 200 km route: d = 195, n = 0, g = 3000 The index for the 100 mi route is: 161 + 3300/44 = 236 km The index for the 200 km route is: 195 + 3000/44 = 263 km Number of Calories burned: 100 mi route: K * 236 = 23 * 236 = 5430 200 km route: K * 263 = 23 * 263 = 6050 One might conclude that the 200 km route was more difficult. One should be aware that difficulty or pain is subjective, and the formulas given above take into account neither steepness of grade nor availability of efficient gearing for a particular grade. --------------------------- Conclusion and Speculations --------------------------- Indexing is useful for quantifying ride difficulty. However, one must be aware of the assumptions above. If a ride consists of relatively steep upgrades followed by gradual downgrades, one will get too much credit for the hills because one might conceivably coast on the downgrades and use very little energy. However, most rides cover a variety of terrain with steep hills and gradual hills. Too much credit earned on gradual hills will tend to be balanced out by credit lost on steep hills. As one's physical condition improves or slackens leading to an increase or a decrease in the normal flat-land riding speed, it may become necessary to re-evaluate D, the cyclist's divisor. To estimate accurately the effect of wind is very difficult. It is generally accepted that for flat loop rides where there is a constant wind vector, the larger that vector the more difficult the overall ride becomes. Since most rides are not ridden under such constant conditions, it is difficult to determine an appropriate handicap for the index of a windy ride. For example, a cleverly planned ride might "catch" the tailwind in one direction with the return trip along a tree-lined road or behind a mountain so that the headwind is blocked or reduced. In the absence of clever planning, a handicap of 5-40% may be reasonable depending on wind speed and typical riding speed. Considering all the assumptions and approximations used in determining a usable index, we have found consistency in our index average speeds (irp) and while-moving index average speeds (mirp) between rides with different amounts of climbing and distance without adding a handicap on windy days. Of course, we plan rides to avoid headwinds as much as possible. There are now several tools on the market that calculate power delivered to the rear wheel. This would be the best way to determine how many Calories are actually delivered to the road. Most everything but body efficiency would be taken into account: hills, rolling resistance, headwind, tailwind, etc. We welcome readers' comments and suggestions. Please send email to the address at the top of this article. ------------------------ Notes ------------------------ The program, bike_power, (by Ken Roberts, roberts@cs.columbia.edu) can be obtained from ftp://draco.acs.uci.edu/pub/rec.bicycles/bike_power.c. ftp'ers are requested to restrict access to 7pm-7am Pacific time. Copyright 1992, Bill Bushnell and Chris Hull. Feel free to distribute this article however you see fit, but please leave the article and this notice intact.