I never knew this existed, but I shouldn't be surprised anymore. There's this event called the Red Bull 400. The idea is to "run" a distance of 400 meters up a very steep incline. Oh, 400 meters isn't that far? True, true. But what if it's 400 meters up an incline at a 37-degree angle? Yeah, that's not so easy.
But just how difficult would it be to "move" up this hill? Yes, I changed it from "run" to "move." I think someone could run part of the way, but not the whole distance. I always go by the running definition where BOTH feet are off the ground at the same time. I'm honestly not sure I could do that on an incline that steep—but maybe someone else could do it.
So, how do you measure the effort required to move up this hill? How about this—suppose I moved up the hill at a nice leisurely pace. Let's say it takes me 30 minutes. That wouldn't be so bad, would it? What would be different with a more race-like time of 5 minutes? Clearly, that wouldn't be so easy. Actually, the most difficult track in the Red Bull 400 circuit is Planica in Slovenia. The record time for that slope is 4 minutes 55 seconds.
Clearly, speed makes a difference. In both the slow and fast run up the hill, you will have the same change in energy. Energy is like money. It's a way for us to keep track of interactions. You can exchange money for goods and services, and an interaction can transfer energy from one system to another. When you climb a hill, you will use some of your own internal energy (stored up by eating Wheaties and Ovaltine) to increase your gravitational potential energy. Here, the gravitational potential energy is defined as:
In this expression, m is the mass of the person (probably you), g is the gravitational field with a value of 9.8 Newtons/kilogram, and Δy is the change in height. With a mass in kilograms and the change in height in meters, you would get a change in energy in units of Joules. As an example: If you take a textbook (about 1 kg) and lift it from the floor to a table, that would take about 10 Joules of energy. So it doesn't matter if you crawl up a hill or sprint. You still have the same change in gravitational potential energy.
Moving up a hill isn't just about energy; it's also about time. You can have a change in gravitational potential energy in a short time or in a long time. With different changes in time, you will have different rates of energy change. This rate of change in energy is called the power, and is defined as:
It's possible that the running human's decrease in internal energy only goes to a change in gravitational potential energy due to the vertical change in position, but it's still best to define the power in terms of ΔE so that we could include other types of energy later. If the change in energy is in Joules and the time is in seconds, you get power in units of Watts. Now for the calculation. What kind of power would a 75 kg human need to output in order to finish the Red Bull 400 in 5 minutes? I'll start with a diagram.
I'm using a 37-degree incline from the Planica hill. If the change in energy is ONLY due to the change in gravitational potential energy, I really only need to calculate the height of the hill (h). Assuming the track is a straight line, then I get a right triangle with a known angle and a known hypotenuse. From this, I get a height of:
Now I can calculate the power based on the change in energy and the time interval (300 seconds).
Umm … damn. That's some serious power, at least for a human. But remember, that's just assuming there is a change in gravitational potential energy. The human also has to do something else during this race—start from a speed of zero m/s and speed up to some running speed. That means they will also have an increase in kinetic energy where the kinetic energy is defined as:
Of course, the runner doesn't have a constant speed during the entire hill climb, but I suspect that the maximum speed is reached fairly quickly. Let's say that the human accelerates to the final running speed instantly. In this case, the speed can be determined from the definition of velocity (in one dimension).
That's not very fast. It's about 3 mph—a typical walking speed (on flat ground, of course). So it's not crazy to imagine that the runner will reach this speed fairly quickly. This means that we can pretty much ignore the "speed up" time in our calculation of the final speed. But what does this do to the power output? Here is the new calculation.
Putting in my values for the Planica run, I get 590 Watts. So the change in kinetic energy part of the power is just a tiny fraction of the total power. It's mostly about running up the hill and not about speeding up. Fine, it's still a huge power output. Just for comparison, a horsepower is 746 Watts. This means a human with a 5 minute run time would need an output of 0.79 Hp. Yup. That's crazy impressive. But here's the cool part. Humans can have superhuman power output even higher than 590 Watts if it's for a very short time interval.
Let's try an experiment. Maybe you can't do this at home, but maybe you can. All you need is a pull-up bar (and enough strength to do a pull-up). In case you don't have a pull-up bar in your house (we have one in a door frame in the hall—the kids love it), here is a diagram of a human doing a pull-up.
Suppose this movement (from hanging to chin at the bar) takes about 1 second and has a vertical distance of 0.5 meters. This increase in height would be a change in gravitational potential energy of 378 Joules. Since the time interval is 1 second, the power would be 378 Watts. I mean, if you can do a pull-up you can probably do a pull-up in 1 second. If you could reduce this to 0.5 seconds, you get a power of 735 Watts—almost one whole horsepower. But even if you could do that, that's just one pull-up. Trying to keep that pace up for 5 whole minutes would be insanely crazy. Insane.
Suppose you were some super powerful human with 1 Hp output for longer than 1 second. Or what if you were more like a typical human with a power output of 100 to 200 Watts for an extended time? What would this do to your Red Bull 400 run time? Let's take the expression above and solve for the time instead of the power. Here's what you get. The only number I'm putting in is the constant 400-meter length of the run—also, I'm ignoring the change in kinetic energy of the runner since it's fairly small.
Now I can put in a bunch of different values for the human power and find the time. This means I can produce the following plot of run time vs. power (all for a 37-degree incline and human mass of 75 kg).
Yes, this is actual python code. If you click the pencil icon you can change stuff. But the important thing to see is that even a super powered human would have a tough time doing this Red Bull 400 run in under 180 seconds (3 minutes) much less the the record time of just under 5 minutes for a very athletic human. In fact, it's even worse than that since I didn't include the change in kinetic energy. This should be more significant for faster times. But I will leave that for you as a homework assignment. See if you can make a better plot of run time vs. power that includes the change in kinetic energy. Or if you want an alternate HW assignment—beat a 5 minute time on the Planica Red Bull 400 track. Good luck with that.