Least strenuous climb
To reach the top of your itinerary you might have the possibility to choose from several routes with different characteristics. Suppose you can choose between the straight and steep one, called A, which arrives directly at the top with the maximum gradient, and a gentler route, called B, longer, with many hairpin bends and modest climbs. Which is the most convenient one? That is to say, which is the less expensive one energy-wise?
To provide an answer, you need to simplify the problem by eliminating all possible variables. Therefore, suppose that the two routes are characterized by the same environmental and soil conditions and that the rider can express the same mechanical power. The latter is the product of the force that pushes the bike-rider system forward and the speed of progress. Even if their product is the same, the individual sizes can be different. In path B, compared to path A, you have a lower thrust force and a higher feed rate, that is, you push less and travel faster.
Let’s ask ourselves a basic question: why do we find it more difficult to ride uphill? The answer is in the concept of energy. Let's start by saying that anything at a certain altitude from the ground, has the so-called "gravitational potential energy" that increases as the altitude increases. Therefore, when the cyclist climbs the mountain he sees his potential energy increase. Since the energy does not come out of thin air, this increase must have a very precise origin, which is the fatigue experienced during pedaling: the chemical energy stored in the rider's body is transformed into the gravitational potential energy of the bike-rider system. It is more difficult to ride uphill because this fatigue is transformed into gravitational potential energy, that is, in a greater altitude from the ground.
If fatigue is connected to the increase in gravitational potential energy, to understand which path is less tiring, we must study its behavior and the parameters on which it depends. At the end of the climb, the increase in gravitational potential energy is equal to the product of the mass of the bike-rider system, gravitational acceleration, and height difference. Lighter cyclists have an advantage going uphill because they spend less energy and it is the same reason that explains the spasmodic research for lighter equipment. Bringing a mass to a certain height is an energy cost that, to save effort, must be reduced as much as possible. The other parameter that defines the increase in potential energy is the height difference and leads us to a first and surprising conclusion: the energy expenditure required by the two paths is the same, because it does not depend on the shape or length of the route, but exclusively on the difference in altitude between the point of arrival and the starting point. Not only that, the other interesting result is that if the energy required by the trails is the same and if the mechanical power developed by the cyclist is the same, then it takes the same time to ride the two paths.
This result, completely derived from simplified hypotheses, has to deal with the reality that always introduces additional variables. When it comes to energy and energy balances, we must always take into account the energy losses caused by friction. They represent another enemy of the cyclist because the effort made to overcome friction is completely lost energy. On path B, the feed rate is higher, as we initially said. Since viscous frictions are closely related to speed, they are greater than those on path A. Think, for example, of the resistance exerted by the air on the bike-rider system. The rolling friction on the wheels as well, linked to the flattening of the tires and therefore to the load on them, is greater in route B. It follows that in reality, the two paths are not equivalent: the less steep and longer one is more energy expensive because it requires overcoming more frictions.
For an even more precise assessment, it is necessary to determine whether the two different thrust regimes are equivalent for the organism, or whether developing a constantly greater torque is more fatiguing, but this does not affect the energy needs of the trail and depends exclusively on the athletic condition of the rider.
If you are interested in the topic, more details can be found by reading "The science of Mountain Bike riding: the physics behind MTB skills" which contains all the topics, addressed to all MTB enthusiasts.
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