Algebra, Energy, Mathematics, Physics, Space, Time

“Pow”! “Bang”! The Physics of Superheroes

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A photograph showing a few frames of a Marvel superheroes comic book. Two masked superheroes - one wearing a red outfit and the other one with a green outfit - are flying through space. The captions read: Red Man "You can open your eyes now. There they are." Green man "Wait. I'm confused. If we're smaller than light particles now. How are we even seeing?" Red Man "You're not... Not in any human way. The five senses become something else entirely at this quantum level. Your mind's doing you a favour. It's processing all this into familiar visuals so you won't go insane. By the way, you're not breathing oxygen either. It's best not to think about it." Green Man "No kidding." Superheroes smaller than photon-life.Physics meets Comic-Book Superheroes

The Physics of Superheroes is a popular science book by  James Kakalios, a Physics professor at the School of Physics and Astronomy of the University of Minnesota, and a long-time comic-book fan.  First published in 2005, the book explores the elementary laws of physics.

The author James Kakalios does not set out to discredit the world of superheroes by showing how it contradicts modern science, granting the heroes one too many “miracle exceptions” from physical laws.  He focuses instead on examples of comic book scenes that can help understand the diverse laws of physics from an unusual angle, such as “Gwen” Stacy’s death… or Ant-Man’s ability to punch his way out of a paper bag! 

Elements of comic books are related to principles of physics, such as momentum, levers and torque, covering a variety of scientific topics from Classical Mechanics to the Quantum World.

Here is an example from what is possibly the coolest Physics textbook ever written.

The question reads as follows:

Batman (mass 80.0 kg) stands on a window ledge 5.0 m above the floor (Fig. 8-40).  Grabbing a rope attached to a chandelier, he swings down to grapple with The Joker (mass 70.0 kg), who is standing directly under the chandelier.  (Assume the stuntman’s centre of mass moves down 5.0 m.  He releases the rope just as he reaches the villain.)

a) With what speed do the entwined foes start to slide across the floor?

b) If the coefficient of kinetic friction of their bodies with the floor is μ = 0.250, how far do they slide?

 

Describing Motion

How fast are Batman and The Joker going to go post-“POW”?

Well, it’s always helpful to draw a diagram about the situation you are trying to visualise, and Kakalios’s book provides a really neat one…

An image from James Kakalios' books of The Physics of Superheroes, featuring Batman and The Joker getting "physical".
Getting Physical with Batman and The Joker

To do this, we must transfer all the gravitational potential energy he has (the energy an object has when suspended above the Earth) to its kinetic energy (the energy bound up in his mass moving at some speed).

The higher you hold an object above the Earth, the more potential energy it has.

This is because the longer it falls to Earth, the faster it goes, and this gives the object more energy.

The gravitational acceleration that causes this potential on Earth, is 9.81 m s-2 (metres per second per second).

The potential energy of an object is

E_P = m g h

So, E_P = (80.0 kg) \times (9.81 ms^{-2}) \times (5.0 m).

At 5.0 metres above the ground, an 80.0 kg Batman (although this figure is unlikely to account for the weight of the superhero armour he is wearing) has 3,924 Joules of potential energy, equivalent to the energy released by the explosion of one gram of TNT!  😮

 

Conservation of Energy

But where does all that energy go once Batman swings back to the Earth’s surface?  According to the First Law of Thermodynamics, the total energy of an isolated system is a constant.  Energy can be transformed from one form to another, but it can be neither created nor destroyed.  The first law of thermodynamics, often known as the Law of Conservation of Energy, recognises the particular concept of ‘internal energy’. 

The internal energy of a system can be changed by heating the system or by doing work on it, the first law of thermodynamics states that the increase in internal energy is equal to the total heat added and work done by the surroundings.  If the system is isolated from its surroundings, its internal energy cannot change.

Remember, in virtue of the Law of Conservation of Energy,

ALL the potential energy becomes kinetic energy.  So E_K = E_P.

 

 

The equation for the kinetic energy of an object is

E_K = \frac {1}{2} \times m \times v^2

Solving the equation for velocity, we have

v = \sqrt \frac {2 E_k}{m}

Hence

v = \sqrt \frac {2 \times (3924 Joules)}{80.0 kg} = 9.90 m s^{-1}

Note. 1 Joule = 1 kg m2 s-2

Batman’s speed when he collides with The Joker is 9.9 metres per second.

For Question a), you need to remember that the momentum of an object is the product of its mass and its velocity.

The equation for the momentum of an object is written as

p = m v

Like energy, momentum is conserved.

If Batman and The Joker entangle during their collision, Batman’s mass and his velocity upon impact must be proportional to the mass and velocity of BOTH Batman AND The Joker considered as one system of objects.

Again, we have a pretty simple equation for the momentum

p_{Batman} = p_{Batman+TheJoker}

Remember momentum p is conserved!

 

Since

p = m_{Batman} \times v = (80.0 kg) \times 9.90 m s^{-1} = 792.0 kg m s^{-1},

we obtain

v_{Batman+The Joker} = \frac {p}{m_{Batman+The Joker}} = \frac {792.0 kg m s^{-1}}{80.0 kg + 70.0 kg} = 5.28 m s^{-1}

 

So, the mass and velocity of Batman during his strike equals the mass of both Batman and The Joker multiplied by the velocity at which they skid across the floor.  With an 80 kg Batman moving at 9.90 metres per second, the 150.0 kg Batman+The Joker pair should begin skidding at roughly 5.3 metres per second.

A heck of a “WALLOP”, if you ask me!!  😮

Answer to Question a)  The entwined archenemies start sliding across the floor at about 5.3 metres per second.  

(“WHIZZZ”?)

But…

How far do the two of them skid?

 

A cartoon showing a roller-skating elephant being pushed into motion by its handler. The caption reads: "Newton's Second Law of Motion".Friction Matters

Question b) involves a bit more Physics, but stick with me.  :-p

We know that the floor offers resistance to the sliding duo, and this is in the form of friction.  During the skid, friction will apply a force to the pair, and that force is equal to the coefficient of friction of the floor (how “rough” the floor is) multiplied by how heavy the object sliding on it is.  Just think, it’s far easier to push a 1 kilogram block across a rough floor than it is to push a 100 kilograms block…

This frictional force is obtained by

F = \mu \times m \times g

So,

F = 0.250 \times 150.0 kg \times 9.81 m s^{-2} = 368 kg m s^{-2}

Note. 1 kg m s-2 = 1 Newton

So, the floor will resist the pair with 368 Newtons of frictional force. 

As many of you already know, force equals mass times acceleration.

F = m a.

The acceleration will result in

a = \frac {F}{m}

And

a = \frac {368 kg m s^{-2}}{150.0} = 2.45 m s^{-2}.

This means that the floor can “accelerate” the pair at a negative rate of – 2.45 metres per second per second (negative because it is acting in the opposite direction that the pair is sliding, so the two superheroes decelerate).

Finally, based on equations derived to calculate the stopping distance for cars, we can figure out that after “SMACK”ing into The Joker, both he and arch-nemesis Batman will slide a significant 5.73 metres.

Answer to Question b)  Batman and The Joker slide 5.73 metres.

Driven to Discover…

This year, Edinburgh’s International Book Festival is getting Stripped and ready with a strand devoted to comic books.  Over 40 events featuring many of the writers and artists behind the comics and graphic novels.  BBC Arts correspondent Pauline McLean has been talking to James Kakalios, the author of The Physics of Superheroes, who describes himself as a “mild-mannered physics professor at the University of Minnesota”…  😉

 

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