As a former Knowles Science Teaching Fellow, I remain committed to high quality math and science education in the United States. I’m particularly interested in math and science education that is grounded in real-life application and develops marketable skills. In my Powered by Physics series, I aim to provide content on the interplay between triathlon and science for students, educators, and triathletes alike. The series is inspired by the enlightening text Bicycling Science by David Gordon Wilson. In this post, I will focus on how an understanding of physics can help you maximize your hill climbing potential.

Putting theory to practice on the hills of the VA-WV border

The force a cyclist applies to the pedals is transmitted through the drivetrain and rear wheel to the ground, propelling her forward. On downhills and flat ground, the main force opposing a cyclist’s motion is air resistance. On uphills however, the majority of the force applied by a cyclist is used to overcome the force of gravity.

Skyline Drive provides ample gravitational resistance

When a cyclist climbs uphill, she does work against the force of gravity, increasing her “positional energy,” which is known in Physics as gravitational potential energy. The higher she rides up a hill, the greater potential she has to accelerate back down the hill. It is equal to the product of the mass of the bike+rider (m), gravitational acceleration (g), and the change in elevation (h). For example, if a 75 kg bike+rider climbs from an elevation of 500 meters to 1,500 meters, her gravitational potential energy increases by

$75\ kg \cdot 9.8\ \frac{m}{s^2} \cdot 1,000\ m = 735,000\ Joules$

The bulk of the work done by our cyclist goes into this increase in potential energy.  However, she must also do work to overcome the force of air resistance, rolling resistance, and bump resistance.¹ Assuming that 75% of the total work done by the cyclist is used to increase her gravitational potential energy, the cyclist does 980,000 Joules of work in the 1,000 meter climb. 980,000 Joules is equivalent to 234 food calories. This may not seem like much, but the human body is at best only 30% efficient in converting chemical energy from food into “useful” energy to power the pedals of a bicycle.² Accordingly, the cyclist in our example will burn at least 780 food calories in the 1,000 meter climb.

According to Chipotle’s nutrition calculator, this burrito bowl has 535 calories, not enough to fuel our rider to the top of the climb!

While the amount of work required to complete a climb is interesting enough, most competitive athletes are more concerned with how fast they can conquer a climb. The time needed to ascend 1,000 meters is dependent on the rate of work the cyclist is capable of producing. By definition, the power output of a cyclist is her rate of doing work. For example, if our cyclist completes the 1,000 meter climb in one hour, her average power is

$Power = \dfrac{Work}{Time} = \dfrac{980,000\ Joules}{3,600\ s} = 272\ Watts$

Again, when climbing, the bulk of the work done is to overcome the force of gravity. As stated before, this is equal to the product m·g·h.  Thus, the power required to overcome the force of gravity while climbing is

$P = \dfrac{W}{T} = \dfrac{m \cdot g \cdot h}{T}$

Rearranging the equation gives

$\dfrac{P}{m \cdot g} = \dfrac{h}{T}$

The left side is the cyclist’s power (P) to weight (m·g) ratio. The right side is the cyclist’s climbing rate, the ratio of height ascended to the time required to complete the climb. This equation tells us that a cyclist’s power to weight ratio is directly proportional to the climbing rate. A 5% increase in a cyclist’s power to weight ratio will yield a 5% increase in her climbing rate.

Armed with this information, it’s clear why the competitive cycling community is focused on power meter based training, deliberate nutrition, and maintaining “race weight.”

Thanks for reading. Time to work on my own power to weight ratio.

Notes:
1. Bicycling Science, David Gordon Wilson – p.123
2. Bicycling Science, David Gordon Wilson – p.37

Key Physics Concepts:
Gravitational Potential Energy (aka energy of position): U = mgh
Power (aka the rate of doing work): P = W/T

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