 ### Determining a Character's Maximum Thrust

Some type of propulsion is required for an object to fly. Wherever this propulsion comes from, SPBs with wingless Flight capabilities can exert some kind of force that propels them through space. An object that exerts a force (in this case, against the mythical ether of the universe) will be accelerated by the equal and opposite reaction force exerted on that object. While flying through an atmosphere this reactive, propulsive force will be counteracted by the frictional forces exerted on the flying object. These frictional forces increase as the object's velocity increases and at some point will be strong enough to exactly cancel the maximum thrust available to the object. At this point there will be no net force acting on the object, meaning it will no longer accelerate; the object is at its maximum possible velocity.

By modeling the strength of this air friction we can determine it's strength at a given velocity for a given object. By evaluating this force at a character's maximum flight speed we can determine the amount of thrust that character can exert. In the velocity range we typically deal with this frictional force will be proportional to the square of the object's velocity. It will also depend on the shape and size of the object, as well as the properties of the atmosphere the object is travelling through. Mathematically, this force can be expressed as:

Ffriction = � d A C v2 = cfriction v2

(magnitude of force due to air friction) = � (density of air) * (cross-sectional area of object) * (drag coefficient, which describes how aerodynamic that object is) * (velocity of that object)2

By plugging in values for the constants used above we obtain the magnitude of air friction as a function of velocity. The density of air at standard temperature and pressure is typically about 1.3 kg/m3; the minimum cross sectional area of a human body (looking down at the person from above) is about 0.25 m2; determining the drag coefficient is very complicated, but values typically fall between 0.1 for highly streamlined objects and can be greater than 1.0 on the opposite end of the spectrum. Very streamlined cars typically have values near 0.25; I will approximate a human value to be about 0.5. This means that for a given power:

Ffriction =  (0.08125 kg/m) vmax2

It seems fair to assume that the maximum velocity given in the HU2 rules refers to horizontal flight. We must therefore add the vertical force exerted on the object by gravity to the horizontal frictional force in order to obtain the thrust required to maintain a horizontal velocity. When the total force on the object is zero and the object is at it's maximum velocity the thrust will be pointing forward and upwards; see the balanced force diagram below: Fthrust2 = Ffriction2 + Fgravity2 Fthrust2 = ( cfriction vmax2 )2 + (m g)2

Rather than keep everything in terms of (m g), the gravitational weight of the character, I will here make an assumption that the standard character weighs about 250 lbs (=1112 N). Feel free to recalculate these figures for your own character's mass because this maximum thrust is going to enter into every flight consideration we obtain. This assumption gives us the maximum thrust as a function of maximum velocity:

Fthrust2 = ( 1112 N )2 + ( ( 0.08125 kg/m ) vmax2 )2

I can now plug in the maximum attainable velocities for flight powers, and obtain the maximum available thrust:

 Type of Propulsion Flight: Wingless; First Level Flight: Wingless; Fifth Level Flight: Wingless; Tenth Level Sonic Flight vmax (mph) 220 300 400 700 Fthrust (N) 1361.7 1836.4 2826.0 8033.7 Fthrust (lbs) 306.1 412.8 635.3 1806.0

What do these values mean? The above amounts of thrust are the maximum possible forces that a character can exert, at any velocity, to counteract any forces that might be acting back on them. This maximum thrust therefore gives us (at very low velocity) the maximum possible weight, including his own, that the character can lift using his Flight powers. Using these values, a 250 lb character with Flight: Wingless can barely lift off of the ground when carrying 50 lbs of added weight, but that same character at tenth level will be able to lift off with almost 400 lbs of added weight.

### Velocity Limits for Additional Weight

While carrying additional weight characters won't be able to reach their maximum velocity. Now that we know the maximum thrust we can work backwards through the same vector math as before, obtaining a maximum velocity for characters carrying additional weight. If that tenth level character carries an additional 200 lbs, his total weight is 450 lbs ( = 2001.7 N ). The maximum thrust he can exert is unchanged, which according to the table above is 635.3 lbs ( = 2826.0 N ), but it won't have as great an effect on the increased mass. Therefore, we can calculate the maximum horizontal speed to be:

Ffriction2 Fthrust2 - Fgravity2
Ffriction2 =  ( ( 0.08125 kg/m) vmax2 )2 = ( 2826.0 N )2 - ( 2001.7 N )2 = 3.9795x106 N2 = ( 1994.9 N )2
vmax2 = ( 1994.9 N ) / (0.08125 kg/m) = 24,552.1 ( m/s )2
vmax = 156.7 m/s = 350 mph

This value is significantly less than the unburdened maximum of 400 mph, but it's not unreasonably slow. This amount of weight seems like an appropriate burden for this character to fly with, if needed. When playing a specific character it might be beneficial to obtain beforehand a few values of maximum velocity while carrying different amounts of weight. If the character wishes to carry a total of 635 lbs, only 0.3 lbs less than his maximum thrust, his maximum velocity will be only 74 mph. This is still pretty fast, but is a great deal less than the usual limit of 400 lbs.