The variables and an equation to consider when starting a physical activity

Posted on October 24, 2016

(Last Updated On: October 24, 2016)

A common scenario I run into with clients is once a given issue -like a hip, shoulder, knee, etc.- has calmed down and is doing well, or after a long layoff, how does one go about returning to previous, or new, activities? Running was problematic for your knee. Now the knee has calmed down and is doing well, or you haven’t run in a long time (or ever), how do you get back into running? What should you consider?

I tend to give the same speech in these scenarios, but it’s not as concrete as I’d like it. I started thinking about the variables I always mention, and whether that could be put into an equation. Despite my mathematics minor, theory was my least favorite part of math classes, and my math is quite rusty. (I also have a limited understanding of any materials science. A lot of engineers read this site. If you have a way to improve this, please share.) Let’s see what we can do.


  1. The given area is actually being used in the given movement.
    1. Using the equation for achilles stress when you’re say, bench pressing, wouldn’t work.
  2. No distinct changes in how you move
    1. If you for some reason run very differently than you squat -even if both involve a bending of the knees- that could change things. We’re assuming form is not an issue at this point, per the intro.
  3. We can’t include, but need to appreciate, everything. That is, “all else being equal,” the equation hopefully holds
    1. Example: You start running, but you also started sitting in a chair for 10 hours a day. The equation will not factor in e.g. how your knees do with the addition of running and sitting so much more. The reality is no equation is perfect. (Not even Newton’s!)
    2. However, you could add multiple stresses together. If it’s your knees you’re concerned about, you could add the addition of your running with the addition of your sitting, giving a generalized knee stress number.

The variables

Strength Intensity

This would be how much force are you putting out. The easiest example is lifting. How much effort are you putting into that lift? 100% being your e.g. one rep max. The closer you are to 100%, the more stress there is.

-> A one rep max chart is relevant. Gist being the more reps you do, the less weight / force you’re engaging with.


Speed Intensity

How fast are you doing the activity? If you’re lifting at only 50% of your max, but doing it as fast as you can, that’s going to be more stressful than lifting at a leisurely pace.

Similarly, while you may be aware of this relationship,


As you move faster you’re not able to contract muscle as forcefully.

That’s not how stress on the body works. That is, stress in the physics sense is proportional to force. However, just because you do something with a very high velocity, and subsequently a lower force, does not mean you’ve decreased the stress on the body. Deadlifting and sprinting both work the hell out of the hamstrings, yet sprinting is where you hear most hamstring pulls occur.

Do something really slow with a lot of force, or something really fast with not a lot of force, OR something with a good amount of force and a good amount of speed, and it’s all stressful to the body.

One of the easiest ways to make a given activity feel better for somebody is to have them do it more slowly.

Maximum speed is how fast you can run for about 70 meters. After that it becomes speed endurance.

notable 100 meter sprint performances splits

After 70 meters even world class sprinters start slowing down.

This is brought up because while some may attempt to do a speed lifting session, like dynamic bench pressing / squatting / deadlifting (/ olympic lifting), and while one may feel like they’re going really fast, the absolute speed of those movements is much slower than an all out sprint. Going back to our force – velocity relationship graph above, the addition of force (weight on the barbell) in these exercises means the velocity has to be less.

-> While trying to find some more info on materials I came across this relevant passage, where we can keep in mind an activity like running is repeated impacts:

“an impact is a high force or shock applied over a short time period when two or more bodies collide. Such a force or acceleration usually has a greater effect than a lower force applied over a proportionally longer period. The effect depends critically on the relative velocity of the bodies to one another.

At normal speeds, during a perfectly inelastic collision, an object struck by a projectile will deform, and this deformation will absorb most or all of the force of the collision […]

However, these deformations and vibrations cannot occur instantaneously. A high-velocity collision (an impact) does not provide sufficient time for these deformations and vibrations to occur. Thus, the struck material behaves as if it were more brittle than it would otherwise be, and the majority of the applied force goes into fracturing the material. Or, another way to look at it is that materials actually are more brittle on short time scales than on long time scales”


The longer you do something the harder it is. Doesn’t matter if it’s laying down or standing in one position. Do it long enough, it gets stressful.

Days per week

The more often you do something, the more stressful it becomes.

-> Caveat: For some, doing something one time per week will be more stressful than doing it two times per week. This is because the body may not really adapt. An everyday example would be a person who only lifts (each body part) one time per week. Some of these people end up sore after every workout, but if they lifted twice per week, the soreness would go away.

I’ve found this incredibly variable, where it’s hard to account for in an equation. Worth keeping in mind though.


The first time you ride a bike in a while, or ever, even if you’re sitting on one of those soft, gel like seats, the next day we’ve all experienced the “Holy-hell-did-I-get-roofied-and-ass-raped-last-night?” sensation.

When something is new, it’s more stressful. This is one of the notable differences between people and inanimate materials. When a fresh metal beam is pushed on in a certain way for the first time, it’s at its most resilient. For humans, we’re at our least. When a metal beam has been pressed “that way” it may begin the process of breaking down “that way.” While for humans, we may break down initially, but then we build back up. The metal’s tolerance begins to decrease; humans it starts to increase.

Number of people involved

Whenever someone wants to get back to a group activity -running / hiking buddies, most sports- I recommend they do the activity by themselves first. The moment others get involved the intensity gets lifted. Competitiveness comes out, trash talk becomes a thing, the mind gets preoccupied with other things more than “Am I going too hard for my first day back?” You play longer and harder than you otherwise would.

This is a fail safe for many who go “Man, I really didn’t think I was playing that hard, but I guess I could have been. Tough when you get into the flow and into the game to really monitor it. Next thing I knew it’d been an hour.”

That is, many will go do something group oriented then tell me they really didn’t go too hard. However, the moment they tell me it was group oriented I know it was two or three or four times more stressful than they say or think.


Stress For Certain Area =

(Strength Intensity^2 + Speed Intensity^2) * Duration^(1/3) * Days Per Week * People Involved * Familiarity

You can get a feel for importance by the math. The intensity variables are exponentials. Small increases in intensity lead to bigger increases in stress.

Duration is cube rooted. That is, it’s significantly minimized. Big increases in duration lead to smaller increases in stress.

->   3^3 = 3 * 3 * 3 = 27.

27^(1/3) = 3.

If we bump 27 by a factor of 2:

27 * 2 = 54.

54^(1/3) = 3.8.

So while the duration has been doubled, it doesn’t represent a doubling in our equation. We went from 3 to only 3.8.

Doing this also provides the ability to compare activities which are durationally very disparate. Think a heavy squat session vs a long run. The squat may only be a couple minutes of stress while the run can be an hour or more. This is how we attempt to equalize some of this.

After that, days per week, people involved and familiarity all get equal footing.


Let’s compare a 5 sets of 5 reps squat session at 80% strength intensity to an hour long run, at 45% speed intensity. Both done two times per week, by yourself, activities you’ve been regularly doing.

Say 5 seconds for every squat rep, totaling 125 seconds for the squat, and 3600 seconds (60 minutes * 60 seconds per minute) for the run:

  • You can see intensity is on a 1-100 basis, not a decimal. The reason for that is exponential decimals get smaller or don’t change. (Intensity of 100% = 1, and 1^2 = 1.) We want intensity values to get bigger, quickly.
  • The amount of people involved is only up to 11, as that’s what the most popular sports will get up to, and that’s likely a fair threshold as to the limit. Whether it’s 11 or 15 people unlikely changes the intensity.
    • Like any equation, this isn’t perfect. Four people in tennis -doubles- may very well be less intense than two people. Many will do doubles because it involves less running. In most activities though, more people means more everything. This is why many do group classes like CrossFit. The extra people gets them to do more than they otherwise would. (One could make a program to account for these differences, but I think we’re getting the point without needing to do that.)
  • For Familiarity it’s someting like a 3 is you’ve never done the activity. 2 is you’ve done it, but still aren’t accustomed to it. Or you’ve done it, but not in a while. 1 is you’ve been doing the activity regularly. (At least multiple days per week, for a couple weeks.)
  • Even if you’re going as fast as possible, you’re still using some strength. Even if you’re using all your strength, you’re still trying to move an object with some speed. So strength and speed can’t be zero.

The equation was set up to make these two types of activities similar, for something like stress on the knee, as in my experience training people a heavy, but not crazy, squat session, would be similar to a long, but not killer, run. I haven’t worked through every possible scenario with it, obviously there are specific situations which wouldn’t fit, and I’m happy to hear some readers give examples of playing with it, but a perfect equation is not the point.

The idea is to consider these variables in this type of context. Where intensity of speed and strength are the first things you want to concern yourself with. Don’t just go full bore into an activity you haven’t done in a while. Don’t do that for a long duration, multiple times per week, with a bunch of people.

From there, tweak these things. Ok, you’re confident you started out at a low intensity, but still had e.g. unusual soreness. How long did you do the activity? Did you rush into doing it every day? How long has it been since you’ve done it?

You swear you took proper precaution…but did you go play racquetball with four other people? You probably did everything longer and harder than you realized.

You want to make your squat session more knee friendly? You can lessen the weight / go slower (granted, going too slow can make lifting hell) / lessen reps / lessen days per week / not be in a group squat session.

If you’d like to play with the spreadsheet, here is a link. If on Google Docs, you can make a copy of it so it’s editable by you. Or you can download an Excel version by clicking the following link, then click the attachment_id link: tissue-stress-activit sheet

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