### B 8. Weight and speed

Figure 1 From pedaling power to cycling speed pedaling power (Ppe), intrinsic resistance (Rb), rolling resistance (Rr),
air resistance (Rd), slope resistance (Rsl) and speed (v)

Pr = m × g × Cr × v
Pd = 0.5 × ρ × A × Cd × v³
Psl = m × g × % × v

Mass
The mass (m) is the sum of the body weight and the weight of the bicycle (both ready-to-ride) plus possible baggage. More mass means that more pedaling power is needed to overcome the rolling resistance (Pr) and slope resistance (Psl). The mass has no role in overcoming the air resistance (Pd) (see the formulas above), unless mass becomes so much larger that it entails an increase of the front surface (A). So a fat belly (if it is mainly at the front) will increase the mass but not the front surface. But genuine obesitas will cause a larger front surface as well as an increase of air resistance.

Touring cyclist
Imagine that our 75 kg non-competitive cyclist were to first lose weight down to 60 kg, then gradually gain weight again to a weight of 90 kg. But his front surface does not notably change during this process. What is the effect of these weight changes on speed when riding the 10 kg racing bicycle and given the same continuous pedaling power of 225 watt?

In the case of a ready-to-ride body weight of 60 kg plus 10 kg bicycle weight (m = 70 kg), the speed on the level road (hands in the drop handlebars, pedaling power 225 watt) is 33.33 km/hour (9.257 m/s):
Pb = 0.05 × 225 = 11.3 watt
Pr = (60 + 10) × 9.81 × 0.006 × 9.257 = 38.1 watt
Pd = 0.5 × 1.23 × 0.4 × 0.9 × 9.257³ = 175.6 watt
Ppe = Pb + Pr + Pd = 225 watt
With a ready-to-ride body weight of 90 kg on the same 10 kg bike (m = 100 kg) a speed of 32.36 km (8.989 m/s) is reached (table 1). So an increase in total weight of 30 kg results in a decrease in speed of 0.97 km/hour or 2.9% (32.36/33.33) at the same pedaling power. The relationship between weight and speed on a level road is linear (figure 2). So 1 kg more mass (total weight) causes a loss of speed less than 0.1% (1/30 × 2.9%) on a level road.

Climbing
Given a ready-to-ride body weight of 60 kg, a 10-kg regular racing bicycle (m = 70 kg) and 225 watt of pedaling power, the climbing speed of this cyclist on a 8% slope is 12.46 km/hour (3.462 m/s):
Pb = 0.05 × 225 = 11.3 watt
Pr = (60 + 10) × 9.81 × 0.006 × 3.462 = 14.3 watt
Pd = 0.5 × 1.23 × 0.4 × 0.9 × 3.462³ = 9.2 watt
Psl = (60 + 10) × 9.81 × 0.08 × 3.462 = 190.2 watt
Ppe = Pb + Pr + Pd + Psl = 225 watt
With a ready-to-ride body weight of 90 kg on the same 10-kg bike (m = 100 kg), the climbing speed on the 8% slope reaches 8.97 km/hour (table 1). So an additional 30 kg of total weight causes 28% loss of speed (8.97/12.46). The relationship between mass and speed is also linear during climbing (figure 2). So 1 kg more total weight means a loss of climbing speed < 1% (1/30 × 28%) on a 8% slope.

Descending
In the case of a ready-to-ride body weight of 60 kg plus 10 kg bicycle weight (m = 70 kg) and the hands in the drop handlebars of the regular racing bicycle, the speed in the descent of the 8% slope (without pedaling or braking) is 54.24 km/hour (15.067 m/s):
Psl = (60 + 10) × 9.81 × 0.08 × 15.067 = 827.7 watt
Pb = 0.01 × 827.7 = 8.3 watt
Pr = (60 + 10) × 9.81 × 0.006 × 15.067 = 62.1 watt
Pd = 0.5 × 1.23 × 0.4 × 0.9 × 15.067³ = 757.3 watt
Psl = Pb + Pr + Pd = 827.7 watt
With a body weight of 90 kg on the same 10 kg bike (m = 100 kg), his speed of descent increases to 64.84 km/hour (table 1). And so an additional 30 kg of total weight, makes him 10.6 km/hour or 19.5% faster on his descent (64.84/54.24). The relationship between mass and speed is also linear during descents (figure 2). So 1 kg more mass (total weight) means an increase in the speed of descent of < 0,7% (1/30 × 19.5%) on a 8% slope.

Table 1. Mass and speed (km/hour) on a regular racing bicycle

 mass 70kg 80kg 85kg 90kg 100kg level road 33.33 32.99 32.84 32.68 32.36 8% uphill 12.46 11.06 10.46 9.91 8.97 8% downhill 54.24 57.99 59.77 61.51 64.84

starting values bold printed Figure 2 The relationship between mass (total ready-to-ride weight) and speed is linear,
on a level road and during climbs and descents

Bicycle
The effect of 1 kg less total weight on speed (2) can be easily tested by leaving behind your water bottle (750 grams), saddle bag with spare tyres and pump (250 grams).

On a level road the average speed of our touring cyclist on his racing bicycle of 9 kg (instead of the foregoing 10 kg) is 32,87 km/hour. His pedaling power (225 watt), cycling position (hands in the curves of the handlebars) and calm weather conditions being unchanged. Previously, with the 10 kg bicycle weight his average speed was 32,84 km/hour (table 1). Thus, with 1 kg less bicycle weight his usual 50 km trainings round takes now 1h 31' 16" in stead of 1h 31' 21". It means that he is back home again 5 seconds earlier (0,1%).

For climbing he rides the 10 km long time trial for touring cyclists on the Alpe d’Huez (10 km long, average slope 8%) (3) twice. The first time, with 10 kg bicycle weight, his average speed is 10.46 km/hour (table 1) and thus his end time is 57' 22". After some rest, he rides the time trial for the second time on the same bicycle, but then the bicycle weight is 9 kg. His average speed is 10.57 km/hour and thus the end time is 56' 46". So 1 kg less bicycle weight renders 36 seconds less climbing time (1%).

For descending (not being urged to brake all the time) he chooses the northern slope of the Mont Ventoux to Malaucène (4). The first 10 km from the top have an average slope of about 8% (5). With 10 kg bike weight his average speed is 59,77 km/hour (table 1); so these 10 km take 10' 2". In the same descent on the same bike, but without water bottle, saddle bag and pump (bike weight 9 kg) he realizes an average speed of 59,42 km/hour; and so these 10 km take 10' 6". That is 4 seconds more (difference < 0,7%).
Thus, the effect of 1 kg less (bicycle) weight on average speed is in fact negligible for a touring cyclist as well on a flat road, as during climb-ing and descending.

Competitive cyclist
A difference in riding time by 5 seconds on a level road over a distance of 50 km (in a time trial, for example) could mean the difference between winning and losing to a competitive cyclist. And a gain in time of 36 seconds when climbing mountains with 8% slopes (being ‘hors catégorie’) also counts, the more because mountain stages usually include more than one mountain. Which is why 1 kg less bicycle weight means a substantial difference for a professional cyclist. However, the UCI has set a limit to the lightness of bicycles: the minimum ‘bare’ weight of a regular racing bicycle allowed to participate in competitions is 6.8 kg (2). In this respect, the recumbent bicycle has a disadvantage compared to a regular racing bicycle: it is much longer than a racing bicycle and, as a result, the minimum weight of a carbon high racer, for example, is at least 2 kg more. The significance of 2 kg more bicycle weight for a professional cyclist, if he might be allowed to participate in the Tour the France on a recumbent bicycle, will be considered in the next chapters.

Conclusions
1. Concerning the effect on the speed, it doesn’t matter whether the weight difference is achieved through the bicycle, the cyclist or the possible baggage.
2. The relationship between mass and speed is linear on a level road, during climbs and on descents.
3. For a 75 kg male on a 10-kg regular racing bicycle, 1 kg less weight will yield a gain in time of 5 seconds (0.1%) at a pedaling power of 225 watt across a distance of 50 km on a level road.
4. With 1 kg less bicycle weight, this male can climb a 10 km long 8% slope at a pedaling power of 225 watt 36 seconds faster (1%).
5. With 1 kg less bicycle weight, this male requires 4 seconds more (0.7%) to descend a 10 km long 8% slope (hands in the drop handlebars).

Sources
1. Wiel van den Broek : Technische artikelen over de fiets: Vermogen en krachten, juni 2013
2. Wikipedia: Alpe d'Huez
3. Altigraph Edition: Atlas des Cols des Alpes volume 1. 1997 I.S.B.N. 2-903968-32-2
4. Wikipedia: Mont Ventoux

Wikipedia.en : Bicycle performance