# How fast can my humanoid run?

The fluff:

I have a humanoid who's about to breach a door. He's got some upgrades to his body and can pump some stimulants through it to boost his strength, resiliance and speed to ridiculous proportions for a short while. When he breaches the door he's going to accelerate from a standstill and run into the room as fast as he can.

The problem:

Many questions have asked about the maximum speed a humanoid can run, but not covered the angle I'm thinking off. When ignoring things like muscle limitations, bone and flesh resiliances etc. You still are limited in your maximum speed. This because if you push off you will eventually start slipping, the surface you are standing on will eventually crumble under the stress you put on it, after every push it takes a moment before gravity pulls you down and your feet touch the ground and air resistance will limit your maximum speed.

The data:

My guy with all his clothing weighs 110 kilo's. He is standing with shoe-rubber on dry, rough concrete. His feet are 23.8cm long and he's 180cm tall.

The constraints:

My guy has a body that can handle anything you throw at it in terms of strength, speed and resiliance. Its his environment that limits him.

The question:

My guy is about to run full-speed from a standstill. His body can handle it, but the environment has its limitations. How fast could my guy achieve maximum speed without destroying his shoes or the concrete he's standing on? What would this maximum speed be?

• I think what will be even more limiting is the acceleration. You can slide very fast on an ice rink, but it is not easy to reach a high speed. A potential answer will probably consider this, and the friction is probably the most limiting factor. Apr 13, 2020 at 13:21
• @B.Brekke yes. That is why I included the rubber vs dry rough concrete, foot size and weight as it will help determine the maximum acceleration possible. But even when going maximum speed you still need to put in the same acceleration forces to counter all the drag forces that prevent you from going faster. So everything revolves around that maximum acceleration. Apr 13, 2020 at 13:41
• I disagree. Maximum speed and maximum acceleration might correlate to some extent, but it is not a coincident that the best 60 meter runners do not even compete in the 200 meters, although you might have a few counter examples. Apr 13, 2020 at 13:51
• @B.Brekke the difference between 60m and 200m runners is biological adaptation. One has more explosive muscle fibers and the other a better mixture of endurance and explosive muscle fibers. But if you look at cars you see that they'll keep accelerating until the dragforces of the air and road are equal to the acceleration forces the engine can put out given that particular gear and how much you press the gas pedal. Since the biological factor is a non-issue for this question it revolves around the maximum acceleration force the person can create to determine speed and acceleration. Apr 13, 2020 at 14:38
• I think what you are basically asking is, is what the yield shear stress of rubber is depending on the normal load and whether that is higher or lower than the friction for "reasonable" normal loads in this case. Apr 13, 2020 at 21:24

## His maximum acceleration is about 9.8 meters per second squared.

The math is really easy.

their maximum accelerating force is equal to the force of kinetic friction, beyond that they are slipping and not gaining any addition acceleration. The force of friction is based on the coefficient of static friction between rubber and dry concrete which ranges from 1.0 - 0.6 depending on source. (it even depends on the roughness of the concrete and type of rubber in the shoes being worn.

Contact surface doesn't actually matter.

Everything else is just math.

F= normal force times mu (coefficient of friction)

Normal force is mass in kg times gravity

(110kg X 9.8) X 1 = ~1078 newtons

A=F/M Acceleration is equal to force in newtons divided by the mass in kg

1078/110 = 9.8 meters per second squared.

Note this is approximately twice the horizontal force generated per second by a elite sprinter on a track.

So assuming soft rubber shoes and dry rough concrete (ideal conditions) and an ideal stride, his maximum possible acceleration is 9.8 meters per second squared. If he is wearing hard rubber boots or it is the smoothed concrete used in many buildings it will be significantly lower.

there is a reason the fastest sprinting records are set with special shoes on rubber tracks, often with a block start. Friction is the limiting factor for their acceleration.

• you hit the nail on the proverbial head. I went for the environment. Even with a perfect grip, the concrete can sustain a limited load. Apr 13, 2020 at 18:18
• The limit for the frictional force is not the same as the limit for acceleration, this is only true for perfect flat non-deformable bodies. A race car, or I think even a sprinter can accelerate faster than 7.8 m/s^2. Apr 13, 2020 at 21:39
• @D.J.Klomp for a runner on a flat surface friction is absolutely the limit, you can't generate any more force to accelerate you beyond this because you have nothing to push off, this is basic Newtonian physics, equal an opposite reaction. A race car has a lot more mass to generate more friction, not to mention spoilers. I even mentioned sprinters, who set records running on surfaces with higher coefficients of friction often using high traction footwear. Rubber to rubber is as high as 1.2
– John
Apr 14, 2020 at 3:41
• This is a good start but my guy isn't pushing a 110Kg guy across the floor, he's pushing off with his legs, gaining air-time and then pushing off again. Each push-off he generates more force than his bodyweight as he's lifting his bodyweight off the floor, but he stops generating force while airborne so you would need to find an average speed based on a full cycle of push-off to the start of the next push-off. Apr 14, 2020 at 7:31
• @John, Amonton's Coulomb friction law is quite a simplification, neglecting deformability and asperities of the material. Besides that when pushing with your shoes against the floor the relative velocity is zero, so you should use static friction. For deformable materials you create an indentation in the material (elastic or even plastic), that you use for acceleration. This deformation is also one of the causes for the difference between static and dynamic friction. Apr 14, 2020 at 8:48

Searching about how much the material can withstand, we get the standard concrete used in buildings. Concrete specs

The office building says 150kg/m2 with 1053kg/m2 for industrial. Your feet are way smaller than a square meter, but let's say you get some fancy technology to distribute your weight around your foot.

You can not apply more than the 1053 kg rated load on the industrial floor without risking collapse.

If your guy is 100kg, again for simplicity, we calculate F = m.a.

Again rounding, 10.000 N = 100kg x a. You get 10 m/s. THen we get that Speed = Initial speed + time x a.
Speed = 0 + 10 secs x 10 m/s. Your guy can clock 100 m/s what means 360 km/hour. Not bad...not bad at all.

• don't forget friction, the 110kg human can only exert 862 newtons of force on the concrete before they start to slip. (mu of rubber to dry concrete is 0.80). Also the OP is asking for acceleration not just top speed.
– John
Apr 13, 2020 at 14:16
• This is perfect for calculating his jump height, but as John said not his forwards acceleration which would be a vector based on the forwards and upwards force of each push-off. As a ballpark estimate to get an idea of the limitations this looks good though. Apr 13, 2020 at 14:41
• This gives un upper end. You can not go faster than the best-case scenario with a perpendicular vector which disregards traction. Go with F1 wheels grip for numbers on the best-case scenario for traction. Yet bear in mind F1 got spoilers to generate more than 2.000 KG downward force. So your tiny puny human-shaped object is going to lack grip. Apr 13, 2020 at 18:16

It depends on the extent your character is allowed to change his body form.

The limit is the air drag vs traction he can get.

Air drag gets equal to the human body weight somewhere 40-50m/s (see "terminal velocity") or, ~150km/h (100mph).

Maximal traction is equal to weight or a bit less. Actually, way less because when running one doesn't touch the ground most of the time (depends on the running technique). It can be more at speed if you manage to get the aerodynamic forces to push you down.

If you are allowed to use/create aerodynamic spoilers like formula 1 cars, you can basicaly get what they can do - ~500km/h / 300mph

• Air-drag gets equal to the human body at 9.8m/s^2 acceleration. So if his acceleration is less than 9.8m/s^2 then his maximum velocity will be less, but at say 18m/s^2 acceleration (if you could) you would run faster than terminal velocity. Although I think that since friction is based on the gravity it'll be the limit unless indeed something like spoilers is used. Apr 15, 2020 at 7:46

It sounds like the issue is one of acceleration, not speed, given that he is running into a room and not the outdoors. Even with a top sprinter, they will hit the far wall long before they get anything like up to top speed, unless it's something like an aircraft hangar or a cathedral.

For better acceleration, he might to better to pull himself along with handholds, or kick off from a rear wall, as friction will be a limiting factor otherwise.

Also, doesn't he have dainty feet for a big guy? :)