Bicycle Power

Enough ranting; back to physics. Today I feel a Cocktail Party Physics-style inspiration from watching the Tour de France. (I love that it is on live every morning with the sprint finish just before lunch. Can get some reading done while it is on in the background, then work on my fall classes in the afternoon.) Towards the bottom of this article, I will give a simple example showing the power required to race up a mountain.

Bicycle racing is about the physics concepts of power and torque. Because these same concepts are important to the design of hybrid cars and in motor racing, I will mention those also.

The quick version is that torque translates into force and is responsible for acceleration (short sprints at the finish) while power is responsible for sustained top speed (whether on the level in a time trial or when climbing).

Physics

This is a bit of an oversimplification because torque and power are not independent of each other: both involve force in their definition. Power is the rate of doing work (work in units of Joules divided by time in units of seconds gives power in units of Watts), and is defined as Work / time, or Force * velocity, or Torque * angular velocity. [Because force is more intuitive than torque to most people, I will use the force version in my examples.] You have to have power in reserve if you want to accelerate while already at high speed (sprints in bicycle racing), but power and force (or torque) can be considered separately in many situations.

(Factoid useful for automobile engines: Power in horsepower = Torque in foot-pounds at an angular velocity of 5252 rpm. Torque and power curves always cross there.)

Two engines (or people) producing the same power can produce a large force, doing a lot of work, in a long time (at a slow speed) ... or a small force, doing a small amount of work, in a short time (at a high speed). Power is about quickness, not brute strength, which is why many sports emphasize "power lifting" training (lifting a given weight 10 or more reps at a time) rather than working on a single lift of a heavier weight. The person who can lift a larger weight than another person in the same time (such as the time needed to sustain a block in football) is more powerful.

Since sport is all about getting there first, power is what usually wins a competition. However, what often matters more is the power to weight ratio, since a heavier person (or car) needs more power to (say) climb a hill at a particular speed.

Tradeoffs

The reason we treat power and force separately (and report both when talking about automobile engines) is that there are physiological and mechanical reasons that the upper limits on power and torque (or force) can vary independently of one another.

Automobiles with a push-rod V-8 have huge amounts of low-end (low rpm) torque. Diesel engines even more so. When you want to pull a trailer or accelerate from a stop light, you need that "grunt" of a large force at a low speed. Hybrid cars get this from the electric motor, which also produces a high torque at low speed. The key design concept of a hybrid is to use the electric motor for torquey things like getting moving, and design the gasoline engine to keep it moving at constant highway speed (where power matters more than torque) as efficiently as possible. Dividing the two functions between two physically separate devices makes the engineering design problem much simpler.

A DOHC (double overhead cam) engine, common on imports and smaller cars as well as high-end race cars, produces the maximum torque at quite high rpm. [My Miata has a torque peak at 5500 rpm.] If you want to get a large starting force from this kind of motor, you need to wind it up (and eat up the clutch) in a way that most people won't do. In my not-so-humble opinion, manufacturers have put bigger motors in "small" cars so that people who don't know how that motor works can get the torque needed to pull into traffic. This comes at the expense of highway mileage because the more powerful motor is less efficient. [My ancient Miata, with its 1.6L engine, got 34 mpg at sustained highway speeds of 77 mph on my last trip. Its top speed of 115 mph is plenty, so it does not need more power.] Again, a hybrid can have a DOHC engine designed only to cruise at 80 mph (100 mph if you are Al Gore) on the highway, letting the electric part deal with acceleration.

Human physiology has similar variability.

Two bike racers with the same mass will exert exactly the same force and do exactly the same work to ride up and over a mountain pass. Same m means same m*g*sin(theta) [force to move against gravity on a slope of angle theta] and same m*g*h [work done to lift you over the pass]. The winner does it faster, which requires more power ... not more force and certainly not more work (or more calories burned). You just have to be able to burn those calories faster, to do the required work faster.

Sprinting, like during the finish of the last few races, is about acceleration. Getting a jump on someone, to increase your speed to a level you can't sustain very long, just long enough to win the race. Competitive riders can get to over 60 mph on a level road, but only for several hundred meters. There is power involved here also, but not sustained power. It is about producing a large force for a short time, and only for a short time. These riders usually cannot climb mountains because they produce extreme amounts of power only in short bursts.

Power still plays a role in sprints, because your top speed occurs when the power you can produce is equal to the power being drained away by friction and air drag. More power translates into a higher top speed, the critical factor in winning an individual event like a time trial. It is also why the finish of a bike race looks a lot like NASCAR, with riders drafting someone until the last moment and then using a "slingshot" move to pass. The most powerful sprinter can only sustain top speed for a few hundred meters, so they time that move to get to the finish line just when they can't do it any more.

Example from today's finish:

I used a stopwatch to time the racers over the last kilometer of Stage 4. They took about 49.5 s, which translates into 20.2 m/s or 45.2 mph. That is the average speed. Since data shown during the previous kilometer indicated the leaders riding at about 34 mph when they started that stretch, the winner (Thor Hushovd of Norway) and the man who almost caught him at the line were probably going in excess of 50 mph as they approached the finish.

I'd really like to know what the computer on his bike said his speed was over the last few hundred meters.

The power demands are huge. The force of air drag increases roughly as the square of the velocity, so the power required increases as the cube of the velocity. You need eight times the power to go twice as fast.

(Side note: This dependence on the cube of the velocity explains why "restrictor plate" NASCAR cars cannot catch up with the draft when coming out of the pits. You asymptotically approach terminal velocity, so it takes more than a lap to pick up those last few mph. You also see this in qualifying, where the second lap on a superspeedway is always faster than the first.)

Example relevant to mountain climbing in the Tour:

The steepness of a road is given in %, which is the slope (rise over run). Thus an 8% grade corresponds to a rise of 8 m in 100 m (horizontal distance), or 100.3 m along the road (the hypotenuse of the triangle). If a bicyclist wants to climb that hill at a constant speed of 15.0 mph (24.1 kph), he has to do the work required to raise his mass (and his bike) 8 m in the time it takes to go 100.3 m at 6.7056 m/s. That work is m*g*h, and it must be done in 14.96 s. The work depends on the mass of the rider, of course.

It is too early to know who will be "King of the Mountain", but some past contenders have masses that range from 61 kg (134 lb) to 73 kg (161 lb). I will use 65 kg as my example, and assume the only other mass involved is the bike (6.8 kg minimum mass). They will also carry some water on longer climbs, because the rules forbid getting any water from a team car during a climb, but nothing extra on short climbs. That value gives us work = (71.8 kg)*(9.807 m/s^2)*(8 m) = 5.63 kJ in 15 s. That is a power output of 376.5 Watts = 0.505 horsepower. Additional power is required to overcome the drag due to the air, the work you need to do when riding that speed on level ground. Thus this is a lower limit on the power needed to climb at that speed.

Note: 1 hp = 746 W, which many top riders can develop for a period of time.

Also note that 1 food Calorie is 4.19 kJ, so they are burning over 5 Calories per minute just overcoming gravity on one of those climbs.

Some bike racing details for wannabe fans:

First, no mention of bike racing would be complete without mentioning the Oscar(R) nominated film, The Triplets of Belleville. Nominated for best animated feature and best original song, it tells a wonderfully strange story of the wildly improbably rescue of a young man who is kidnapped while in a bike race. That said ...

Mountain climbing starts this weekend with Stage 7 on Saturday. The page I linked to defaults to the route, but you can also view the "profile" of the route (showing the elevation changes) and the "passes", which gives the length and grade of each major climb.

Climbs were classified from easiest (4) to hardest (1) before anyone thought someone would be crazy enough to race over ones that are harder than the hardest. (Classification of whitewater rapids uses an open-ended scale with 1 as easiest to avoid this problem.) The H category (haute or high) has since been added to indicate climbs that are basically impossible for normal human beings. Difficulty is a combination of steepness and length, so it is possible for a very long 6% climb (such as the ones out of Val d'Isere on Tuesday's Stage 9) to be rated H while a short climb at 8% (like near the start of Stage 17) would be only a 3.

The really interesting climbs are on days with an H category (or two) in the route. Those include Stage 9 (linked above) on Tuesday, July 17, with two big climbs, Stage 14 on Sunday, 22 July, with two (one is at the finish so a climber will win that stage), Stage 15, with one on 23 July, and Stage 16, on 25 July, with two (again, one at the finish). Stages 14 and 16 may determine who wins the overall title.