How to find height by measuring speed

Thinking about Physics while riding a bike

(From Night Café — prompt by the author)

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What better opportunity to solve a physics problem than to be along a mountain, on a bike that slides down a slope without pedaling? To tell the truth, I happened to look at the bike’s display right before getting off, and as soon as I reached the bottom, I wasn’t expecting such a steep descent! The idea of turning it into an experiment came later (I stopped to write down the data on my phone… and then processed it a few months later!).

The question was: “Damn, how high was this descent?” The only data available are the initial speed at the top (v₁) and the final speed (v₂) at the bottom. What I want to know is the height I came down from; with just this data, I have to trust to one of the most important principles of Physics: the…

Conservation of energy

That is: the total energy of an isolated system is conserved over time. For now, we are talking only about mechanical energy.

The system, composed of me and my bike, initially has two energy components: one due to its motion (kinetic energy K₁) and one due to its position, high up from the bottom of the descent (potential energy U₁). The total mechanical energy of the system is

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When we get to the bottom of the descent, we can no longer descend further, so we no longer have potential energy (U₂ = 0), but we still have kinetic energy K₂. If the total energy needs to be the same, I can equal the initial and final values:

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This tells us that potential energy has transformed into kinetic energy. Recalling the formulas of kinetic energy and gravitational potential energy:

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where g is the acceleration of gravity, and m is the mass, which appears in all terms, so it can be simplified (Galileo was right to say that the mass of a falling body does not count!). Now I solve with respect to h:

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The height does not really depend on the difference in speeds, but on the difference in their squares: this amplifies the effect! (4–2 is very different from 16–4…). Now I translate the data into the right SI units (m/s) and insert them into the formula, obtaining a height of 2,88 m. It may seem a small value, but psychologically, 3 horizontal meters seems less than 3 vertical meters… (which corresponds to a flight of stairs…).

The simple problem ends here, as I reached what I was looking for.

What about friction?

I actually made a big simplification: as I was going down, I felt the wind on my face, so there was friction with the air. Friction is a dissipative force: part of the total energy is transformed into heat (I can’t feel it because, as soon as it’s generated, it’s carried away by the air itself, but my face, bike, etc, would warm up).

Result: the measured final kinetic energy is less than the frictionless one (that is, the frictionless final velocity must be greater than the measured one); consequently, the actual height must also be greater than the calculated 2.88 meters.

But since it is a force that does not conserve energy, how can I use the conservation of energy? Simply by introducing a term that takes into account the lost energy!

I then have to subtract the value of this energy; but how much is it worth?

Well, the calculation is a bit complex, but let’s see how I can get an evaluation without getting lost in the mathematics.

I have to suppose that there was no wind (I remember it was exactly like that); the speed is high enough to create air vortices (and you can feel them!), so I have to use the quadratic dependence on the speed: the friction force is:

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The coefficient k is something like:

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• A is the area of ​​the body + bike that makes air resistance;
• ⍴ = 1,2 kg/m³ is the air density;
• Ca is a dimensionless aerodynamic coefficient, that depends on the body shape and material.

From some specialized cycling sites, I find that a typical k value for an upright cyclist position is around 0.45–0.55 kg/m. In fact, the body area is around 0.8–1.0 m²; the aerodynamic coefficient is 0.5 for a smooth sphere: since I’m neither round nor smooth…, I can estimate 0.8–0.9, considering the bike’s structure. Using these values, I get 0.55 kg/m, in line with what I found online. I decide to use an intermediate value: 0.50 kg/m.

To find the dissipated energy, I have to calculate the work done by this force, which always has the opposite direction to the velocity, therefore it’s a negative work:

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where s is the displacement during the descent. However, the velocity changes as the position changes (it’s an accelerated motion!):

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I must then add all the small works for every section of the path; in practice, I am introducing an integral (as usual for work):

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where D is the total length of the path. The problem is that v(s) is the solution of a differential equation… Let’s try to go on with an approximation: we could try use a constant value for the speed; for example, the average. We have to average a squared speed, so we could use:

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In this case, the work is simply

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As I could imagine, the work depends on the path length. However, I didn’t measure D; by eye (and by memory…), I recall it was around 12–15 meters. To check that it’s a coherent number, I’ll try to calculate the corresponding slope.

Relation between height, angle , and length

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That is about 12°, and this corresponds to a slope of around 20%. This is a steep slope, but I remember that it was for this very reason that I thought of calculating the height. So the assumption is consistent: I enter the data and calculate the dissipated energy:

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This is the amount of energy that must be removed from the total energy. The conservation equation becomes:

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Now I cannot eliminate the mass of me and my e-bike: I evaluate this to be 100 kg. By solving with respect to h’, I obtain:

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The absolute value was inserted just to underline that L is negative, so the final result is an increase in the height by the term

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So, with friction, the estimate goes from 2.9 to 3.3 meters, a correction of 14%.

A check

I have to check if the approximations made make sense. The differential equation to obtain the speed as a function of space (using the law of dynamics, F = ma)

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has a solution like the following (I used the very same data values):

v(s) — x: position, y: speed (by Author)

We can see that

• the increase in speed decreases with higher distances: the speed tends to a constant value (the speed limit);
• this increase is rather constant within a reasonable length: it made sense to consider the squared speed average.

For a different range of speed values (such as a car), the importance of friction would be different. When the aerodynamic coefficient becomes larger (that is, more friction), the speed reaches its limit soon, and we must handle the approximations with care.

Now I have to confirm the hypothesis by calculating the speed limit in my case; this is done by equating the friction force to the gravitational force along the slope:

Weigth component along the slope equals friction force (by Author)

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The measured speeds are very far from this value, so the approximations are valid.

In summary

I first applied the conservation of energy to the simple task of finding the height of the slope. Then I tried to refine the measurement by introducing air resistance. I evaluated the values that were not available, trying to get coherent data and checking that they did not contradict the assumptions. In this way, I arrived at an approximate but consistent result.

Physicists are often accused of living in a frictionless world; I have shown that physics can treat friction as necessary. But if we introduce friction from the start, it can hide fundamental concepts and important laws describing reality. Example: Newton could not have stated the law of inertia if he had considered friction.

Secondly, often the effect of friction is not so high. For instance, if we are going to calculate the speed limit of a falling body, well, friction is the main actor. But if we are talking about a heavy ball rolling down a slope, the friction effect could be as low as 10% or lower.

I wish everyone many wonderful bike rides: when you hit a slope, now you know what to do!