This looks like something out of a video game, but it's for real. Some parkour masters can scale a building by jumping back and forth between two facing vertical surfaces, zigzagging upward. Seriously, check it out right now, I'll wait. (It's the move that starts at 0:10 in the clip.)
I'm not about to attempt this, because my job as a physics professor would be in jeopardy if I fell on my head. But from the safety of my office chair, I can walk you through how it works, because parkour is all about using physics to your advantage.
Use the Force, Luke
Friction is your friend here. When two surfaces are in contact and able to slide against each other—like shoes on a wall—we can model the resistance to sliding as a frictional force. The strength of this force depends on two things:
1) The specific materials in contact. We describe this interaction with a coefficient of friction, represented by μ. It's generally a number between 0 and 1. The coefficient for rubber soles on stone is higher than the one for, say, smooth leather soles, so don't try this in your dress shoes.
2) How hard they're pushed together. We call this the normal force, since it's perpendicular (i.e., “normal”) to the stationary surface. Imagine trying to slide a book on a table while some joker is pushing down on it. Same thing here, only sideways: As you leap onto a wall, your foot pushes into the wall. The frictional force Ff is then the normal force N times the coefficient μ:
Why do we need friction? Well, unless you are in a video game, you need an upward force to counteract gravity, so you don't just plummet to the ground. And if you want to actually move upward, this frictional force has to exceed the gravitational force.
Now, if there is a force pushing on an object, that object will accelerate—meaning its velocity will change. If there's more than one force, we add them up to get the net force. Newton's second law says the net force on an object equals the product of its mass (m) and acceleration (a). (Or more usefully, acceleration is net force divided by mass.)
Both the force and the acceleration are vectors—that's what the arrows over those symbols indicate. It just means they have a direction as well as a magnitude.
Jumping Onto a Wall
So you're a parkour newbie and you want to try to run up a vertical wall. You get a good running start and leap, getting a foot on the wall. Hold it! Freeze time right there. Here's a diagram of your foot (as a block) at that moment:
There are three forces acting on your foot (and so the rest of you). First, there's the gravitational force (mg) pulling you down. If your foot doesn't slide right off, the upward-pushing friction force (Ff ) must equal the gravitational force, so the net force in the vertical direction is zero.
Finally, we have a force from the wall pushing on the foot. Wait, what? Let me explain: As you hit the wall, your foot pushes inward. That impact force is what creates the momentary friction. Now this might seem like a silly question, but it's fundamental: Why don't you crash through the wall?
It's because the wall pushes back on you—that's our normal force N. Newton's third law says, for every action, there is an equal and opposite reaction—meaning a force of equal magnitude, in the opposite direction. Forces always come in pairs.
But wait! Only one of these opposing horizontal forces is acting upon you. That means your foot will accelerate away from the wall—and once you lose contact, there's no friction to hold you up.
Zigging and Zagging
OK, the wall is repelling you back to the right, so what's your next move? Use that rightward motion to cross to a second wall several feet away! And to move upward, you want to push up as well as forward before you lose contact.
As you hit the second wall, your foot once again pushes inward, creating friction and a reaction from the normal force. Then you can turn that rebound into another leap back to the left, and so on. If your timing is right, you can keep zigzagging upward like the guy in the video.
So let's try this with some real numbers, shall we? Here's what a single jump looks like:
We can break that diagonal motion down into horizontal and vertical portions; for now let's just focus on the former. Say you start with a horizontal velocity (v1) of –1 meter per second and rebound with a horizontal velocity (v2) of +1 m/s. The change of sign indicates the reversal of direction. Think of it like you're moving back and forth along the x axis of a coordinate plane, negative to the left, positive to the right.
Notice that your speed stays the same, but the velocity changes. (Remember, velocity has a direction.) In fact, because your horizontal velocity reverses, you get a big increase in velocity. (v2 – v1) = (1 – (–1)) = 2. This gives you a larger impact acceleration, a greater normal force, and more friction. The bouncing back and forth is the whole key to beating gravity in this stunt.
So how much force would you need to exert to make one of these rebounding wall jumps? Let's say you have a mass of 75 kilograms and a friction coefficient of 0.6, which is probably conservative for rubber soles.
For starters, the frictional force (Ff) must equal or exceed the gravitational force (mg). The gravitational field strength on Earth (g) is 9.8 newtons per kilogram. So the gravitational force, (m x g) = 75 x 9.8 = 735 newtons.
Now remember, the frictional force is the normal force times the coefficient of friction (Ff = μN). So to achieve a minimum frictional force of 735 newtons, we need a normal force of at least 1,225 newtons (Ff/μ = 735/0.6 = 1,225).
Both of these forces, gravity and the normal force, are pushing on you, so we need to add them up to get the net force. Since they're perpendicular, we can easily calculate the vector sum as 1,429 newtons. (Take note, kids: You want to be a parkour hero? Take linear algebra.)
That means you need to push back with the same force (because forces are an interaction between two things). 1,429 newtons is a force of 321 pounds. That's significant but not impossible. Doing it eight times in rapid succession, though? Not so easy.
How much time do you have to do the turnaround? With the normal force and mass of the person, we can calculate the horizontal acceleration ax. By definition, that in turn equals the change in velocity per unit of time (Δt), so we can use that to solve for the time interval:
Plugging in our numbers, we get a time interval of 0.12 second. In other words, if you hesitate you fall. Bottom line, if you want to do this awesome parkour stunt you gotta be strong, fast, and fearless—because if you run short of newtons halfway up, the descent is a lot faster than the ascent.
