How to Run Up a Wall—With Physics!

March 20, 2019 | Story | No Comments

I can't decide if this looks like something from a super hero movie or from a video game. In this compilation video of crazy stunts, a guy somehow finds a way to bound up between two walls by jumping from one to the other. "Somehow," of course, means with physics: This move is based on the momentum principle and friction. Could you pull it off? Probably not. But you can at least do the math.

The first key to this move is the momentum principle, where momentum is the product of an object's mass and velocity. The momentum principle says that in order to have a change in momentum, you need a net force acting on an object. This can be described with the following vector equation. Note that everyone uses the symbol "p" to represent momentum.

How about a quick example to show how this works? Take an object like a pencil, a ball, or a sandwich and hold it out at arm's length. Now let go of the object. After your hand releases contact with the object (in my mind, it's a sandwich), there is only one force acting—the gravitational force pulling down. What does this force do to the object? It changes the object's momentum in the direction of the force. So after 0.1 seconds, the object's momentum will be in the downward direction which means it speeds up (since the mass is constant). After the next 0.1 second, the object gets even faster. In fact, the sandwich will continue to speed up as it falls until there is another force acting on it (from the floor) to slow the sandwich down. Don't worry, you can still eat it if you get it before the five second rule is over.

The second key idea needed to run up a wall is friction. Friction is a force that acts on an object when two surfaces are pushed together. For a fairly reliable model of this force, we can say that the friction force is parallel to the surfaces interacting with a magnitude that is proportional to the force with which these two surfaces are pushed together. This would be modeled as the following equation:

This expression is for the maximum friction force between two surfaces. In the equation, μ_s is the coefficient of static friction that depends on the two materials interacting and N is the force pushing the two surfaces together (we call this the normal force).

So you can see that this friction will be necessary to run up the wall. However, you can't just run up a vertical wall because there would be no (or very little) normal force between your foot and the wall. With no normal force, there is no friction. That's bad. You fall.

Now for the video. The guy is able to run up the wall by first moving towards, then moving away from the wall. This means that there is a change in momentum (since it changed direction) which causes a force to change this momentum. In this case, the force is from the wall. The problem is that you can only push on the wall for a short amount of time before you move away from it and lose contact. What makes it work here is that there is another wall on the other side so that the runner can then switch and repeat the move again.

Here is a diagram, this should help.

If I can estimate the change in momentum and the time interval for one of these wall jumps, I can calculate the force from the wall and then the required frictional force. Let's do it.

For this analysis, I will need to get an approximate position of the human in each frame and I can do this with video analysis. I'm making some guesses on the size of things, but I suspect it's close enough for a rough value of friction. Although the camera zooms and pans during the motion, I can correct for that with some software. Here's what it looks like.

But that's not what I want. I want to look at the change in momentum. Here is a look at just the x-motion during this wall climb.

From the slope of this position-time graph, I can get the x-velocity before the collision with the wall with a value of 1.39 m/s. After the collision, the dude is moving with an x-velocity of -2.23 m/s. If I assume a human mass of 75 kg, this is change in x-momentum of -271.5 kg*m/s. Also looking at the graph, I get an interaction time of about 0.2 seconds. The change in x-momentum divided by the time interval gives the average x-force with a value of 1357.5 Newtons (in the negative x-direction). For the imperial people, that is a force of 305 pounds. Yes, that's a lot—but it's just for a short period.

Since this force is in the x-direction, it is the same as the normal force that pushes between the foot and the wall. Using the model of friction above, I can solve for the coefficient of friction since the magnitude of friction must be the weight (75 kg x 9.8 N/kg = 735 N). This means the minimum coefficient of static friction must be 0.54 (there are no units for this coefficient). And that's just fine. This table of coefficients of friction lists rubber on concrete with a range of 0.6 to 0.85—so this is entirely plausible.

Still, I wouldn't recommend you try any old "plausible" move. It takes a long while to train from plausible to possible.