# What’s Your Ideal Running Speed to Conserve Energy?

It’s easier to stay in shape than it is to get in shape. That’s why I often go for a run. My teenage son usually joins me. It’s great time to raise interesting questions. Here’s one he came up with.

What speed would he have to run to have to lowest total energy usage? Would it be better to go fast and get it over with or would it be better to go slow?

Great question. My initial answer was that running fast will require more energy (and obviously power) because you have more air resistance and must increase the speed of your legs (or you will fall down and go boom). But it also seems that going slow also will waste energy. If you go at a snail’s pace, you are going to spend more time doing things like breathing and thinking and blinking.

So, there probably is some optimal speed that gets a human from point A to point B with the lowest amount of energy consumption. Here is the fun part: I’m going to calculate this optimal running speed based on my own estimates. I’m not going to Google stuff.

### A Model For Energy Consumption.

Let’s start with a running human. I want to get an expression for the amount of energy this human uses. I can think of three things the human needs to do while running:

- Push against the air—assuming some air resistance force.
- Increase the kinetic energy of the legs during each stride—and do it again for the next stride.
- Breath and perform other normal human functions—like circulating blood and thinking.

I can get an expression for the contribution to energy use for these three aspects and tuse that to find the optimal running speed.

Let me start with drag. Suppose the human has to push against the air (which is true). I can use the following model for the magnitude of the air drag force.

In this model, the air resistance force is proportional to the square of the velocity. I can find this constant k by considering a skydiver falling at terminal velocity. In this case the air resistance is equal to the weight of the jumper. Assuming a mass of about 75 kg and a terminal velocity of 120 mph (54 m/s), I get a drag coefficient of 0.25 N*s^{2}/m^{2}.

Notice that I didn’t really look up anything here. I just assumed the air drag on a runner is similar to the air drag on a skydiver. Oh, sure, I knew the approximate terminal velocity of a jumper, but that was it.

Now let’s say this runner must expend energy to beat this air drag. That means the runner has to “exert” this same force that the air drag pushes. From this, I can get an expression for the work done over some distance (d)—this is also the energy required by the human for this air drag.

Next up, the legs. OK, this is a bit tougher—but clearly that won’t stop me. Let me consider the energy during one stride (moving both legs) with a stride length of s. A human has to start with a leg on the ground and increase the speed of this leg so that it can be placed in front of the body. I’m not sure if this is the best estimate, but I am going to say this leg has to increase from zero m/s to the speed of the runner. Over the length of a complete run cycle, both legs have to be increased in speed (and thus kinetic energy). Oh, but the legs stop—so does the human get this energy back? Alas, no. That’s not how humans function. So, this is the energy required for 1 stride.

Now for the whole run, I would need to multiply the energy per stride by the number of strides. Again, using a stride length of s and a run distance of d, I get the total energy spent in moving the legs as:

The last part of the energy required to run is the most basic breathing stuff. I am going to say that a human uses energy at a constant rate. This means I can first write this part of the energy as a function of time and then as a function of distance and velocity.

I don’t know why, but I picked this proportionality constant as ‘q’. But now I can get an expression for the total energy by adding these three energy terms. Again, this is the energy required to run some distance (d) at a velocity (v).

Remember, this is my own model based on my own assumptions and estimations. I offer that reminder before you complain about something.

### Estimating Constants

In the energy equation above, there are some variables that need estimates. Let me start with the constant q. How much energy does a human use to simply be alive? Suppose that I laid around doing nothing all day. How much energy would I need to consume so that I didn’t lose or gain weight? My completely ballpark estimation is 500 calories (food calories) or about 2 Mega Joules (MJ). If I use this energy evenly over 24 hours, then the q value would be 23.1 J/s.

For the moving legs term, I need to estimate the mass of one leg and the stride length. From my previous investigation into mass of body parts, I found that a leg has a mass about 0.155 times the mass of the human. That would put each leg at 11.6 kg. For just a rough approximation of the stride length, I am going to use 2 meters.

Oh, I already have an estimated value for the drag coefficient, k. But still, remember that these estimates are for a particular human. A different human would have a different stride length and leg mass and stuff.

### Optimal Running Speed

I will start with a plot. Suppose a human runs 5 kilometers. What is the estimated total energy for different running speeds? I will calculate this for speeds from walking (1 m/s) to Olympic record running (about 6.5 m/s).

From this graph, you see one important thing—slower is better. But of course you could have guessed that. Everyone knows it is easier to walk a 5k than to run it. Sure, it will take longer, but you don’t save any energy by going faster. Oh, if you zoom in you can see that 1 m/s is NOT the lowest energy. There is a minimum energy around 1.2 m/s.

But there is another way to find this minimum energy speed? Yes, this is the classic max-min problem from your calculus course. The basic idea is that you can take the derivative of this energy function with respect to the velocity variable. This derivative will give you the slope of the graph at any given velocity value. When the energy graph is at a maximum or minimum, the slope will be zero. So, just take the derivative and set this to zero. Solve for the velocity that produces this zero slope and then look to see if it is a max or min value.

Let’s do it. Here is the derivative.

Surprisingly (at least to me), the running distance cancels. Putting in the other values, I get a minimum energy speed of 1.24 m/s or 2.77 mph. That’s a brisk walk. That’s how fast you should move—for everything.

Some final comments:

- I’m rather surprised that the air drag term and the moving legs term in the energy expression were proportional to velocity squared. For some reason I expected the moving-legs term to be linear with velocity.
- The distance term dropping out bothers me just a little. I wonder if I did something wrong.
- One thing I did correct is units. You can look at each term and check to make sure it has units of Joules.
- How could you get a better model? Well, you could of course look at what others have done—but I like to start from scratch. If you wanted an experimental method, you could measure the amount of carbon dioxide exhaled by a runner at different speeds. From this you should be able to get energy as a function of velocity and then find the minimum.
- Technically, I skipped a step in my max-min problem. I should check to make sure it’s actually a minimum value and not a maximum.

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