### 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 (weight total) 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 and thus increase air resistance as well.

Calculations
Imagine if 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 regular 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
A speed of 32.36 km (8.989 m/s) is reached with a ready-to-ride body weight of 90 kg on the same 10 kg bike (m = 100 kg) (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 the 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.

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

Mass = body weight + bicycle weight; A = 0.4 m²; Ppe = 225 watt
starting values belonging to a body weight of 75 kg 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

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 bicycle (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 the speed is also linear during climbing (figure 2). So 1 kg more total weight means a 0.93% loss of speed (1/30 × 28%) on a 8% slope.

Descending
With a mass of 70 kg (60 + 10) 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
At a mass of 100 kg (90 + 10), his speed of descent increases to 64.84 km (table 1). And so an additional 30 kg of total weight, makes him 10.6 km/hour or 19.5% faster in his descent (64.84/54.24). The relationship between mass and speed is also linear during descents (figure 2). So 1 kg more total weight means an increase in the speed of descent of 0.65% (1/30 × 19.5%) on a 8% slope.

Bicycle weight
When purchasing a new racing bicycle, 1 kg less bicycle weight can entail a difference in price of over one thousand euro. But how much faster can you actually go with 1 kg less bicycle weight?
On a level road, our 75-kg non-competitive cyclist on his 10-kg regular racing bike reaches a speed of 32.84 km/hour thanks to his 225 watt pedaling power (table 1, bold print). With an empty water bottle (-750 g) and with no saddle bag and pump (-250 g), the ready-to-ride weight of the regular racing bicycle is 9 kg. At the same pedaling power of 225 watt, his speed on the level road will be now 32.87 km/hour. With that, he arrives home 5 seconds sooner after a level road training round of 50 km (50/32.87 versus 50/32.84). Thus for a non-competitive cyclist, 1 kg less bicycle weight means in fact nothing on a level road. However, you can pull up (accelerate) faster, which enables you to close a gap in a group of cyclists more quickly.
During a climb on a 8% slope, this cyclist on a 10 kg regular racing bike reaches a speed of 10.46 km/hour (table 1, bold print) With that, a 10 km long climb takes 0.9560 hours (10/10.46) or 57 min 22 sec. With 1 kg less bike weight, his speed is 10.57 km/hour and he needs 0.9461 hours or 56 min 46 sec for a 10 km long climb. That is a difference of 36 seconds.
On his descent (with his hands in the drop handlebars), he reaches a speed of 59.77 km/hour (table 1, bold print). A 10 km descent then takes 0.1673 hours (10/59.77) or 10 min 2 sec. With a bicycle weight of 9 kg, his speed of descent reaches 59.42 km/hour. The 10-km long descent on the lighter bicycle then takes 0.1683 hours (10/59.42) or 10 min 6 sec. And so an additional 4 seconds are needed in the descent as a result of the 1 kg less bicycle weight.
When climbing and descending, the advantage of 1 kg less bicycle weight is 32 seconds (36-4) on a 10 km long 8% slope. This is a gain in time of 0.8% on a total time of 1.1233 hours (0.9560 + 0.1673). Whether or not this gain justifies a substantial investment in a lighter bicycle is a question that each individual non-competitive cyclist must answer for him/herself.

Competitive cyclist
A gain in time of 5 seconds on a level road (in a time trial of 50 km, for example) could mean the difference between winning and losing to a competitive cyclist. And a gain in time of 32 seconds on a ‘Hors Catégorie’ climb would mean a substantial difference in the outcome of a mountain stage, usually including more than one mountain. Which is why competitive cyclists want a racing bicycle that is as light as possible. However, the UCI has set a limit in this respect: the minimum ‘bare’ weight of a regular racing bicycle allowed for racing is 6.8 kg. In this respect, the recumbent racing bicycle has a disadvantage: it is longer than a regular racing bicycle and, as a result, the minimum weight of a recumbent high racer, for example, is 9 kg. The significance of this difference in bicycle weight when participating in the Tour de France, will be considered in the last chapter.

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 (0.65%) to descend a 10-km long 8% slope (hands in the drop handlebars).

Source
1. Wiel van den Broek (2017): Technische artikelen over de fiets: Vermogen en krachten