# How do you equate a “stair run” to traditional running in terms of Power?

I want to be able to convert the power required to do one exercise, specifically running down then up four flights of stairs, into terms of a run of distance x in time y.

I've found several work and power calculators online, but none of them include stair runs.

Here are the variables:

• Mass: 205 pounds (93 kg)
• Height (4 flights): 40 feet (12.2 m)
• Angle of descent/ascent: 30°
• Based on the height and angle this equates to 80 feet of total distance traveled each way (the hypotenuse), and 69.28 feet of purely horizontal distance traveled each way.
• Time: 30 seconds

How would you roughly translate this into the following format?

I ran x distance in y seconds.

I'd appreciate either a general or specific solution.

• Please remind me to have a look at your question some time this week, I'm sure we can come up with some kinematical estimations – Ivo Flipse Nov 29 '11 at 0:28
• @IvoFlipse Here's your reminder :) – hobodave Dec 2 '11 at 22:23

To answer this question I'll have to make some simplifications and assumptions, because the real kinematics are rather sophisticated.

We'll calculate the Potential Energy, which is the amount of energy required to move a mass up a certain height against the gravitational pull:

E-potential (in Joule) = mass (kg) * gravitational acceleration (m/s2) * height(m)

which equates to: 93 * 9.8 * 12.2 = 11,119.08 Joule. While this seems a lot, 1 kcal = 4186 Joule, so you'll this would only cost 2.6 kcal.

Power (Watt) = Energy (Joule) / Time (seconds)

So taking the 12,000 J / 30 seconds = 370 Watt

Other points to take into consideration:

• the equation calculates the energy and power needed to elevate your mass 12.2 meter, to do so obviously means you'll have to move your body. Given the efficiency of your body is probably between 20-30%, you'll actually have to burn up to 5 times the amount of required energy.
• when running you sway your legs forward, lifting them up in order to swing them efficiently. If you're running upstairs, the required forces around the joints of your leg will be higher, because you're swinging them up against the gravity and lifting them higher than during regular running. These forces are ignored in the equation, but the faster you run the higher the required forces will be.
• swinging your arms help accelerate your body, but obviously that comes at a cost. Like swinging your legs, swinging your arms requires additional energy, which are ignored.

You may be wondering, doesn't it matter how fast I'm running up the stairs? Well given the energy to elevate your body stays constant, so we have to at least use that amount of energy. For this we would calculate the Kinetic Energy:

E-kinetic (in Joule) = 0.5 * mass (kg) * velocity (m/s) * velocity (m/s)

given that we know the required energy we can estimate the velocity in meter per second. We take the total amount of energy (11,119) and divide it by 30 seconds to get the amount of potential energy / second = ~370 J. Then we take the square root of the energy per second divided by 0.5 * mass, which gives us 2.82 meter per second. This equates to ~10 km/h, which is quite the pace.

Off course you could run up the stairs faster, but this is just a calculation based on how much energy would be required to go up. You'd also need to take into account that moving horizontally for a given distance costs you energy. But as I said, I've been making a lot of simplifications and only calculate the required speed and power.

Though as you can see, running up a flight of stairs equates to about 370 Watt, which is a very respectable workout. But obviously, this is only maintained for 30 seconds and I doubt you'd be able to keep it up for long stretches of time.

• First we assume @hobodave is a point mass... – Matt Chan Dec 12 '11 at 3:15
• A point of mass doesn't have arms and legs to swing ;P – Ivo Flipse Dec 12 '11 at 6:24