_{Batman and Superman. [Source: Wikimedia Commons] - CC BY 2.0}

When you look at how the superheroes appear in their flying attires, you would always wish you could be given a moment to be in their shoes. The Batman and Superman's attire with an extra piece of cloth extending makes it more appealing to the eyes.

Looking at how Batman glides with his scallop-hem cape and how Superman can leap tall buildings with a single bound, I began to wonder if it could be the cape doing the magic for it seems to be a common part of the costume in quite a number of superheroes' attire. This led me to research the possibility of the cape having an aerodynamic advantage in their flight.

But guess what, I found out that the superheroes capes might actually be a death machine if they are to be considered as an aerodynamic tool.

Aerodynamically, Superman will actually have his cape to blame for a reduced flight speed while Batman, after enjoying the cape-assisted flight, would probably meet his death from hitting the ground with an impact comparable to being hit by a car at a high speed attributed to the contributions of his scallop-hem cape.

Hope I haven't left you confused about what Aerodynamics is or why each of these Superheroes whom we are craving to be like would witness such calamities after they had been successful with winning our hearts?

In simple terms, Aerodynamics simply has to do with the motion of an object against a gas, mostly air. So, this gives an idea of why this term, aerodynamics is important and why it interests us to see how it relate to our superheroes.

But, before we go deeper, it is necessary to get familiar with the aerodynamics of flight and skydiving which will aid the understanding of the reason why we concluded that the capes give our Superheroes their bossy look could be a suicide machine.

In our case here, we are to analyze the forces that lead to flight as well as those that act on a body falling from the sky, whether it is a skydiver using a Parachute or our favorite Batman gliding down in his trademark scallop-hem cape in the name of saving us from the bad guys!

In achieving flight, four forces are always in play; the thrust, Drag, Lift, and Weight. Our major player here is the lift whose net magnitude results in how fast the object is projected into the sky.

Going back to the basics, if you take an object and throw it up in the sky, sooner or later, we find it returning back to the ground owing to the popular force gravity which is courtesy of the earth pulling everything towards itself. So, basically, the idea of flight or descending has always being to negate gravity either positively or negatively.

What are the forces of Flight

So, going back to our forces, we have them in pairs, those acting horizontally - the thrust and drag, and those acting vertically - the lift and weight. As said, gravity is the brainer here so it is best to identify it first and see how others play along with it.

Weight is the component of gravity gotten from the product of the mass of the object and force of gravity. So, for a body to achieve flight, it has to overcome the weight which is a factor of its mass as gravity is considered constant throughout the surface of the earth (a little difference occurs due to earth's imperfect spherical shape).

The lift is the opposite force to weight which is usually dependent on the net magnitude of the horizontal forces, drag, and thrust. So, we need to reach a magnitude of lift greater than that of the weight to achieve upward movement.

Thrust and drag are more aerodynamically inclined as they are the ones that have to do with the movement of the object in the air. Drag or air resistance is the force that prevents the body from moving in its desired direction. It is depending on the surface area of the object in contact with air and the density of the air in that region. So, the thrust is the work input by the body in other overcome the drag and surpass it to achieve the forward movement.

So, in summary, thrust aims to overcome drag (or air resistance) hence creating a net force the create a forward movement. This net force will be so great that it gives the body the needed vertical lift to overcome the weight and Viola! we are up, flying!.

We, however, need to be clear that Superman might achieve his flight in a different way is being Superhero, but scientifically, he must have created a thrust greater than the drag and hence, a lift that supersedes his weight.

Unlike the wingsuit which adds to the aerodynamic lift (or air resistance) of a skydiver, the Superman's cape rather adds more drag to his flight.

Take a flag flapping from being his by the air as an instance, a drag force is generated as the air molecules hit it. The cape which is left flapping as Superman flies does the same, reducing his lift and adding to his weight.

The drag produced on a body moving against a fluid is calculated using:

D = ^(𝜌C_dV2A)/₂

Let's say Superman's cape is 50cm by 180cm and he was moving at a velocity, V say 25m/s with the air density, 𝜌 being 1.225kg/m³ and coefficient of drag on the cape, C_d taken as 1.1, the drag generated by this cape would be:

D = ^{(1.225 x 1.1 x 252 x 0.5 x 1.8)}/₂
D = 379N (about 38kg)

This value, 38kg is already more than half of Superman's weight which definitely would give the gravity more hand. So, if Superman wasn't fictitious, he probably would have to do away with this cape so as to avoid a reduced flight.

Well, for a better flight, it would be best if the Superman's cape is made to look like the wing-suit usually adopted by skydivers.

Instead of allowing the cape float in the sky, flapping, he could add to his lift by tieing the rear edge of the cape to his feet and restrain the middle from bulging out probably by fixing it to his waist.

This way, we would achieve something close to the wing-suit, hence, adding to his lift even though it wouldn't contribute as much lift as airplane wings do but it at least support flight rather than the suicide antics of the floating cape giving our Superman his hero looks!

Let's move on to talk about Batman's Predicament.

This is where it gets interesting considering that our man decides to answer gravity's call - descending with the help of his bat-like scallop-hem cape.

Here, we are more concerned about how the big effect of gravity is countered by a piece of clothing aiming to add to the air resistance offered against his safe landing (or simply gliding down the sky).

What are our major forces here? How do they work relatively? What does the cape contribute to these?

The physics behind a falling object is majorly determined by the force of gravity gladly pulling it along with its mass.

So, when a body jumps down a cliff or high buildings like Batman does, he is expected to move at a velocity comparable to that of a racing car at its highest speed.

The major force that will hinder the sole control of the motion by gravity will be the air resistance (or drag) trying to prevent the weight from falling through it.

In the case of Batman, this air resistance ought to be increased by the additional resistance offered by his cape whose surface area traps in more air molecules and thus offers an opposition to the motion solely anchored by gravity. But as we have mentioned, it still doesn't offer the countering force that would enhance a safe landing scientifically.

Now, we will need to discuss some terms which will make us understand why Batman cape doesn't offer enough protection for Batman.

At the start of the descent of a body from a height, the body moves faster as it travels downward, this is because the gravity continues to pull the body towards the surface of the earth. The motion of this body is described by Newton's second law of motion for which:

F = ma

hence

a = ^F/_m

In the equation above, a is the acceleration of the body and m is the mass of the body while the force, F is the net force from the difference in the weight of the body and the Drag (air resistance) offered by the atmospheric air. Thus, the acceleration can be expressed as:

a = ^{(W - D)}/_m

The drag offered in form of air resistance has a direct relationship with the square of the descent velocity, V of the body i. e. D ~ V². So, as the velocity increases, the drag is increased in squares of the velocity and hence, it wouldn't be long before this air resistance equals the body weight.

At that point where the drag equals the weight descent velocity, acceleration becomes zero, i.e.

a = ^{(W - D)}/_m <=> a = ⁰/_m = 0.

The body from here then moves with a constant velocity as described by Newton's first law of motion which says that a body continues in a uniform motion unless it is compelled by a force which tends to change the motion. This uniform velocity is termed the Terminal velocity

On reaching this terminal velocity, the motion is now independent of the weight nor the air resistance but moves solely at this high velocity till it reaches the ground, maybe to crash.

So, in simpler words, a body descending first owes its speed of descent to the force of gravity until a point is reached where the opposing force offered by the aerodynamic environment equals the weight and the acceleration becomes zero. Then, the remaining part of the journey is owed to the terminal velocity attained when this equilibrium of force is reached.

Now, let's do some validation.

If Batman, weighing about 100kg with his attire and weapons included, descends a skyscraper of height 150m. Taking as 1.1 and air density as 1.225kg/m³, we can estimate the terminal velocity at which the batman will land if his cape has a surface area of 5m². We have that:

D = ^(𝜌C_dV2A)/₂
Thus,
V_t = √^(2D)/_𝜌CdA

Considering the fact that the cape will offer an additional lift to Batman's flight, he'll glide to a height above the 150m starting height away from the skyscraper. When he descends, this cape also offers an additional air resistance as he falls, and thus, we could say that he'll reach terminal velocity at a point say 120m to the ground. This terminal velocity exists when D = W (Drag is equal body weight), so:

V_t = √^(2W)/_𝜌CdA
V_t = √^{(2 x 100 x 10)}/_{1.225 x 1.1 x 5}
V_t = 17m/s

At 17m/s, Batman ideally shouldn't be controlled by his cape but rather fall to the ground hitting it at that massive speed. But then, being fictitious, he is still able to glide and overturn himself, landing in grand style.

So, again, we can see why the cape is nothing but an additional piece of clothing to Batman's costume.

We would need a larger cloth, shaped like a Parachute which will offer a more significant drag that reduces the terminal velocity to the barest minimum that would allow a safe landing.

The Batman's cape is thus a suicide machine for it doesn't prevent the batman from reaching a terminal velocity that will offer safe land. So, scientifically, to avert this and reach a safer terminal velocity, we need a bigger cape, shaped and worn in a way that it offers something similar to the design of a Parachute or the wingsuit often used by the skydivers.

The aerodynamics of flight is dependent on four forces, namely the thrust, drag, lift and weight (force due to gravity). For a flight to be achieved, we need to generate a thrust larger than the drag offered by the aerodynamic environment thus leaving us with a lift force that is greater than the weight of the body.

The main target of flight is to negate the effect of weight. With Superman's cape allowed to float in the air, an additional drag is generated by it as it is hit by the air causing more drag force to be added hence helping weight get a more upper hand.

Superman if not "super", needs a cape which stays restricted from flapping and attached in a way that it offers to lift as in wingsuits rather than contributing more drag to the system.

The aerodynamics of a falling body with respect to the air resistance needs an object whose surface area contributes a large drag that would be greater than the force of gravity pulling the body to the surface of the earth.

So, the descent velocity needs to be overhauled by this "bullish" drag force contributed by the object. In our case, we would favor a bigger and parachute designed cape for the real-life Batman so as to avoid reaching an outrageous Terminal Velocity to would see him hit the ground like a crashing aircraft.

Superheroes defy the laws of physics and only survive because they are fictitious. If I were to fly like Superman does, I will first need to do away with his floating cape before I think about finding means of generating the needed thrust to propel me into the sky. The cape only generates additional drag which is detrimental to the aim of achieving flight.

Equally, Batman in his trademark scallop-hem cape would hit the ground like a crashing plane if he was human like the skydivers. He would need the help of a bigger piece of cloth as seen in parachutes.

The invention of Parachute for skydivers evolved over the years from the wooden frame parachute invented by Louis-Sébastien Lenormand to the modern ones which are seeing use in many areas other than safe landing which was the originating reason for its invention.

As discussed in a post by @greenrun, we now seek for better ways of saving commercial passengers from a crash incident by means of individual parachutes and probably a case of detachable aircraft that can be safely landed by a parachute. Equally, talks about safe landing space shuttles by means of parachutes are in progress.

So far we can achieve a reduced terminal velocity, landing with parachute could see more and more applications in coming years.

Thanks for reading till the end.

References

• Johnson D. Anderson and Sci=ott Eberhardt, 2000 - "Understanding Flight: An Intro. to Aerodynamics". McGraw Hill Series in Aeronautical and Aerospace Engineering
• Johnson D. Anderson, 1991 - "Fundamentals of Aerodynamics". McGraw Hill Series in Aeronautical and Aerospace Engineering
• T. Tolay, 1975 - "Introduction to Aerodynamics of Flight". [NASA SP-367]
• Are Superhero Capes Aerodynamic? by Daniel Kolitz - Gizmodo | GizAsks
• "Physics Shows Batman's Cape Is Suicide Machine" - Wired
• "Free Fall and Air Resistance" - PhysicsClassroom

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