Compression Strength of Bone and Brick: Sin City

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Abstract

The purpose of this project is to compute the compression strength of bone and brick using two scenes from the 2005 movie, Sin City.

Movie Synopsis

Sin City is a compliment of four inter-connected tales of death, corruption and sex in the fictitious Basin City. The four interlocked tales include of some Frank Miller's best such as "The Yellow Bastard part 1 & 2," "The Hard Goodbye," "The Customer," and "The Big, Fat Kill." The film Sin City has a large cast of stars from all over the place from Bruce Willis to Jessica Alba and with such a large cast comes a diverse set of characters for the film.

Some of the films main characters are the following:

John Hartigan: a 60-year-old detective with a busted heart trying to stop a molester but great misfortune falls upon him along the way.
Nancy Callahan: the 11 year-old-girl who is saved by John Hartigan and later on falls in love with him.
Marv: a brutish thug who goes on a crusade to find his lover's killer.
Dwight: an associate with the old town girls who goes on a mission to cover up a cop killing but has trouble along the way.

Graphic novel and comic writer Frank Miller wrote the Sin City stories before they were made into a film. Elizabeth Avellan, Frank Miller and Robert Rodriguez produced the movie and directed by Frank Miller and Robert Rodriguez. This movie is filmed in a unique way, being one of the few movies that were shot entirely on a "digital backlot". The process required the actor to act with a green screen behind them and have special effects added electronically during post-production. This particular film was shot to be more like a comic book coming to life. Another movie which used this technique was the recent 300.

Analysis

Scene 1

The first scene we analyzed was when Jackie Boy tries to shoot Dwight but the gun backfires making the chamber shoot into his own skull. We wanted to see if the gun had enough strength to break through Jackie Boy's skull.

We first found the time it took for the chamber to travel from Jackie Boy's hand to his skull by finding how many frames it took over the amount of frames per second:

(16 frames)/(25 frames/second) = 0.8 s

Then we measured the distance between an average human arm to his skull and it was 0.8 m. We found the average velocity of the gun with the following formula:

v_f = 2 v_avg = 2 ∆s/∆t)
v_f = 2 (0.8 m)/(0.8 s)
v_f = 2 m/s

The speed we found seems a bit slow but it's the movies and what can you do.

Then we found the average acceleration given that the gun barrel came to a complete stop in one frame.

a = Δv/Δt
a = (2 m/s)/(1/25 s)
a = 50 m/s²

We used Newton's second law of motion to find the force of the barrel assuming its mass was half a kilogram.

F = ma
F = (0.5 kg)(50 m/s²)
F = 25 N

We computed the cross-sectional area of the barrel …

A = πr²
A = π (0.019 m)²
A = 0.00113 m²

and used that to find the pressure.

P = F/A
P = (25N)/(0.00113 m²)
P = 22,000 Pa or 22 kPa

The amount of pressure that is required to break crush bone is 170 MPa (Cameron, et al.) and the amount of pressure we calculated was not enough. A measly 22 Pa would only dent your skin. Ouch!

Scene 2

The second scene was between Marv and the hit man, where Marv was held at gun point. Marv using his monstrous strength and fighting ability shoots one hit man twice and then smashes the other hit man's face into a brick wall, visibly crushing the bricks. We wanted to see if the whether the brick will break first or the skull.

We first found the time it took for the head travel and break through the brick by counting he number of frames it took over the amount of frames per second:

(17 frames)/(25 frames per second) = 0.68 seconds

We know that the mass of an average human being is about 5 kg (Gekhman). The head is displaced through an arms length and the average human arm length is 0.8 m. Using the same methods as in the previous scene, we found the impact velocity …

v_f = 2 v_avg = 2 ∆s/∆t)
v_f = 2 (0.8 m)/(0.68 s)
v_f = 2.35 m/s

acceleration …

a = Δv/Δt
a = (2.35 m/s)/(1/25 s)
a = 58.8 m/s²

and force …

F = ma
F = (58.8 m/s²)(5 kg)
F = 294 N

We discovered that the average human head is about 0.18 meters long and 0.1 meters wide. Thus the area of a human head would be

(0.18 m)(0.1 m) = 0.018 m²

We found the pressure of the human head with the following formula:

P = F/A
P = 294 N/0.018 m²
P = 16,000 Pa or 16 kPa

The compression strength constant of a brick is 11 MPa (Integrated Publishing). 16 kPa would be enough to cause a lot of pain and maybe even a concussion but not enough to break either the bricks or the poor thug's skull.

Conclusion

In both scenes the objects (the gun barrel in the first scene and the head in the second scene) are traveling to slow to crush either bone or brick. The scenes look awesome but they could only happen in Sin City.

Sources of Error

We assumed that the mass of the head of the thug would be the same as the mass of an average human head. However this could be different
We also assumed that the arm length of Marv and Jackie Boy is the same as average arm length.
We also assumed that the distance between the point where Marv grabs the thugs head to where it smashes the wall is one arm length.

Bibliography

Gekhman, Dmitriy. Mass of a Human Head. The Physics Factbook. 2006.
Integrated Publishing. Strength of Brick Masonry. Builder 3 & 2, Volume 01. 2007.
Cameron, John R.; Skofronick, James G.; Grant, Roderick M. Physics of the Body. Medical Physics Publishing: Madison, Wisconsin. 1999.

Allen Ma and Imran Akhtar -- 2005

Physics on Film