Vy= (V0 * Sin(Theta)) - gt

Vx= V0 * Cos(Theta)

Now to Derive Formulas Needed...

Vertical Velocity

Vy = (V0 * Sin(Theta)) - gt

At the heighest point Vy = 0

So: Vy = 0

and :

Vy = Vo * Sin(Theta) - gt

so:

0 = Vo Sin(Theta) - gt

Solved for t

t = (Vo * Sin(Theta))/g

However, this is time to Max Height

Total time in air = (2Vo * Sin(Theta))/g

Inputting this into normal range formula

Range= ((2Vo^2) * Sin(Theta) * Cos(Theta))/g

Therefore, R = 2V0^2 * Sin^2(Theta)/g

**A Song of Ice and Fire: Physics Analysis**

**By Lord Kyle Camacho and Ser Justice Centrello**

What is a Song of Ice and Fire?

A fantasy book series by George R.R. Martin

Takes place in another world during a medieval-esque era (but with dragons)

Features many chracters with a lot of depth and intriguing story arcs/plot twists

Physics in Scene

Formulas

Scene Chosen

This scene occurs in Season 3 of the television show based on the book series

It is a funeral tradition of House Tully to light the funeral boat on fire with a flaming arrow

Edmure Tully, the eldest son of Hoster Tully (the man in the funeral boat), fails to hit the funeral boat after three attempts; his uncle, Brynden "Blackfish" Tully, takes the bow from him out of frustration and manages to hit the funeral boat on his first shot

The Sigil of House Tully is presented to the left.

The topic in physics covered here is projectile motion

We will analyze the physics in this scene and then give it an appropriate physics rating

Video Clip

Formulas - Horizontal Direction

Vx = Vox

Dx = Voxt

Finding Needed Values

Formulas - Vertical Direction

Vy = Voy + gt

Dy = Voyt + 1/2gt^2

Dy = (V^2y - Vo^2y/2g)

Dymax = Vo^2y/2[g]

[g] = 9.8^m/s^2

Edmure's Shot

Finding Bow Initial Velocity

Using Data from Bow Lee's Bows, bow most like the one used within the Clip is the "Palmer"

Has an average FPS ( feet per second) of 170 - which equates to ~52 meters/second

Finding Angle of Shots

Using GIMP Software, the angles of both Edmure's shot and Bryndyn's Shots can be obtained

Angle = ~41 degrees

Angle = ~ 44 Degrees

Bryden's Shot

Quick Analysis

Conclusion

Since Both Bryden and Edmure are physically fit, it is assumed they both pulled the bow to its maxium tension, and in turn had the same Inital Velocity

Range = ((2Vo^2) * Sin(Theta) * Cos(Theta))/g

R = (2 * (52 m/s ^2) * Sin(41) * Cos ( 41))/ 9.81 m/s

R = (2 * (2704 m^2/s^2) * Sin(41) * Cos ( 41))/ 9.81 m/s

R = (5408 m^2/s^2) * Sin(41) * Cos ( 41))/ 9.81 m/s

R = (5408 m^2/s^2) * Sin(41) * Cos ( 41))/ 9.81 m/s

R = (5408 m^2/s^2) * .6561* .7547)/ 9.81 m/s

R = (2678 m^2/s^2)/ 9.81 m/s

Range = 273.0 Meters

Range = ((2Vo^2) * Sin(Theta) * Cos(Theta))/g

R = (2 * (52 m/s ^2) * Sin(44) * Cos ( 44))/ 9.81 m/s

R = (2 * (2704 m^2/s^2) * Sin(44) * Cos ( 44))/ 9.81 m/s

R = (5408 m^2/s^2) * Sin(44) * Cos ( 44))/ 9.81 m/s

R = (5408 m^2/s^2) * Sin(44) * Cos ( 44))/ 9.81 m/s

R = (5408 m^2/s^2) * .6947* .7193)/ 9.81 m/s

R = (2702 m^2/s^2)/ 9.81 m/s

Range = 275.5 Meters

Finding Actual Distance to Boat ( x)

In Chapter 58 of the Book

A Clash of Kings

it states...

"They could not dress more than twenty ships abreast, lest they risk tangling their oars and colliding with each other."

The width of a galley is about 10 feet wide, meaning the river is most likely 200 feet wide, or 60.96 meters

. By looking at the picture on the next frame it can be seen that the boat is nearly directly in the middle of the river so its X distance is equal to 30.48 Meters

Finding Distance to Boat ( y)

Using the picture, it appears that the funeral boat is approximately 100 meters away, or slightly longer than a football field

Actual Distance to Boat

Use the Pythagorean Theorem

Hypotenuse = Sqrt( a^2 + b^2)

D = Sqrt ( 30.48^2 + 100^2)

D = Sqrt ( 929.0+ 10,000)

D = Sqrt (10,929)

D = 104.5 Meters

The range of Edmure's shot was 273.0 meters

The range of Brynden's shot was 275.5 meters

The estimated distance the funeral boat was from them was 104.05 meters

Simply Put, both men overshot the funeral boat by about the same distance

In order to use the Range equation, we needed to find a) the angle in which the shots were fired and b) the initial velocity of the shots. The angles were found using GIMP software. The initial velocity was obtained from Bow Lee's Bows. Assuming that both men pulled the bowstring to its maximum tension, the arrow would have been released at ~52 m/s.

Both men overshot the funeral boat by approximately the same distance, since they both shot the arrow at about the same angle and both had the same initial velocity. Hitting a boat that distance with an arrow is difficult for even trained professionals anyway. In a world filled with White Walkers, warlocks, red priests, and draagons, you can't expect it to be too realistic. The final verdict for this particular scene is...

PGP. Pretty good physics in general. Yes, there may be flaws, but it isn't completely unrealistic in hitting the boat with an arrow. If their initial velocities were decreased or if their angles changed, they would have hit the boat if they were aiming correctly.

**Sources**

GIMP 2.8.10,The GIMP team, www.gimp.org, 1997-2014, retrieved on 31.07.2014.

Martin, George R. R. A Game of Thrones. New York: Bantam, 1996. Print.

Martin, George R. R. A Storm of Swords. New York: Bantam, 2000. Print.

"Walk of Punishment." Martin, George R. R.. Game of Thrones. HBO, . 14 Apr. 2013. Television.

"The Science behind a Bob Lee Bow." Bob Lee Archery/Wing Archery. Web. 22 May 2015.