Formula for determining maximum flight speed of superpowered individuals

Question

I would like to be able to roughly determine the maximum horizontal flight speed of any superpowered individual using only the maximum acceleration that person can produce. To do this, I will need a formula for which I can simply enter the maximum acceleration in g's, or maxiumum force in Newtons, and then get a velocity out.

How flight works:

The kind of flight relevant to this questions works by accelerating the individual's body in a given direction, as if their body were being pulled in a specific direction by a gravitational field of a certain strength. A flying person's top speed is determined only by the maximum acceleration they can achieve, and their interactions with the atmosphere. If they were to fly in space, they could continue to accelerate indefinitely. In other words, there is no hard limit to the speed of flight, except for that imposed by air resistance.

For Example

Someone who can output a maximum of 2g of acceleration could fly at a top speed of roughly 240 km/h. This is because the terminal velocity of a person is roughly 240 km/h, and this flying individual could output 1g of acceleration upwards in order to counteract gravity and hover, and then another 1g sideways in order to fly in that direction. However, I doubt I can simply multiply this speed by a person's sideways acceleration to get a new top speed. E.g: I doubt someone producing 2g of sideways acceleration would travel at 480 km/h, and 3g at 760 km/h, etc.

Assumptions:

• For the sake of simplifying the equations, the person is a sphere with a radius of 0.5m, and weighs 100kg.
• The person is flying at sea level.
• 1g of acceleration is used to maintain altitude with the rest being used for forward movement
• The person will continue accelerating until drag equals the force they produce with their acceleration.
• The person is strong and durable enough to be unaffected by any adverse affects from flying at great speed.

What I Am Looking For in an Answer

A good answer will provide a formula which will output a speed dependant upon the force or acceleration, and hopefully an explanation of all the relevant variables of the formula.

I am not sure if this is necessary, but perhaps it would be useful to split the answer into 2 separate formulae, one for subsonic speeds and one for supersonic speeds.

While I am looking for hard answers using equations and known science, I am not using the hard-science tag, as I know how people get when you use that with questions relating to magic or superpowers.

Why I am Asking

Since this kind of flight has a relatively rigid definition of how it functions, it would be useful to be able to roughly determine any superheroes' max flight speed based on how much weight they could hold and hover with. For example, if one supe is capable of hovering while lifting a car above their head, they must be capable of producing at least 16gs of acceleration (and ~16,000 Newtons) upwards (100kg bodyweight, 1500kg car), and necessarily must be able to fly faster than someone only capable of hovering while holding another person. Being able to estimate their flight speeds is then useful for powerscaling and planning conflicts.

Answer 1

Unfortunately, there isn't going to be any simple answer to this question. The best you're going to be able to get is a general range, since drag depends upon the shape and attitude of the object that is moving through the air, as well as its surface properties. A superhero in slick, rigid body armour flying head-first through the air would have less drag than one who is effectively nude (since flesh is distorted and displaced by rapidly-moving air, and is less smooth), and much less than one who is wearing loose clothing. Similarly, attitude is important, as flying in a vertical attitude (i.e. belly-forward) increases drag when compared to a head-first attitude.

Now, given that there are variables out of our control, we can calculate an approximate speed for a human form, given a particular amount of thrust. If we start with the speeds of skydivers, where a belly-down position results in a terminal velocity of around 200kph, a head-down position results in a terminal velocity of 240-290kph, and minimising drag and optimising body position allows speeds of up to 530 kph. As this is taking place on Earth in approximately 1 ATM of pressure (from 4000m to 1700m above ground level), we know that the force on the falling object is 1g.

From the Wikipedia entry for Speed Skydiving, we can get the formula for terminal velocity:

$$v_t=\sqrt{\frac{2mg}{\rho AC_d}}$$

where:

• $m$ is the mass of the object ($kg$),
• $g$ is the acceleration ($ms^{-2}$)
• $C_d$ is the coefficient of drag (Dimensionless)
• $\rho$ is the density of the atmosphere ($kgm^{-3}$)
• $A$ is the Projected Area ($m^2$)

Special helmets and body suits are used in speed skydiving to minimise $C_d$, which ranges from about 0.7 head-first to 1.0 belly-first. $A$ for a human body is about $0.18 m^2$ head-first, or $0.7 m^2$ belly-first. $\rho$ can be approximated to $1kgm^{-3}$ at 2200m altitude, but is $1.23kgm^{-3}$ at sea level.

So, to use the OP's example of 2g (discounting any forces used to maintain altitude) and a mass of 90kg at 2200m altitude, we can plug in the numbers and get about 71m/s (256 kph) belly-first or 167m/s (602 kph) head-first.

However, as speeds increase toward the speed of sound, we must then consider transonic drag. As speeds approach Mach 0.8, the coefficient of drag increases significantly, peaking at Mach 1, and reducing thereafter, though not entirely back to its subsonic value. This means that speeds generated by the formula above will not be valid for speeds in excess of around 270 m/s without adjustment of $C_d$ to allow for transonic drag. Calculating transonic drag is non-trivial, and appears to depend heavily on the shape of the object. However, a $C_d$ increasing to 4-5 times that of the subsonic (Mach < 0.8) value at Mach 1 appears to be reasonable. For speeds > Mach 1, wave drag $C_d$ proportional to $\frac{1}{\sqrt{Mach^2-1}}$ is expected.

Since $C_d$ changes over the range of transonic and supersonic speeds, it would be necessary to calculate speeds in a tool such as MS Excel using recursive calculation.