Tuesday, November 26, 2013

The Force of Tension of Katniss Everdeen's Bow

Recently the second movie in the Hunger Games triology, The Hunger Games: Catching Fire, was released.  Many of you have probably seen it already, but I would like to preface this entry by saying that it may contain spoilers for those of you who have yet to see the movie or read the book.

I am going to talk about the scene at the end where Katniss shoots an arrow charged with a bolt of lighting into the force field barrier to destroy it.  I would like to determine what the tension force of the string in the bow is to allow Katniss to shoot the arrow into the force field.  In order to preform these calculations I need to make some initial estimates and assumptions:

Bow:
Mass of arrow = 0.5 kg
Change in the angle between the sting at rest and taught = 30˚
The bow doesn’t bend when the string is pulled
Katniss shoots the arrow straight up
When arrow is knocked it bisects the string
It takes 2 seconds to hit the force field

Arena:
Radius = 1000 m
Katniss is 50 meters from the edge of the arena when she shoots the arrow

Given these assumptions we need to determine the distance the arrow needs to travel:

d2=1000m2 - 950m2
d =312 meters

Now we can determine the average velocity of the arrow

vavg = d/t = 176 meters/2 seconds = 166 m/s

Since gravity is acting the opposite direction of the velocity of the arrow then we know that the instantaneous velocity is equal to

166 m/s + 9.8 m/s2 (1 second) = 176 m/s

Now that we know the initial velocity of the arrow, we can determine the tension force acting on the arrow using the equation:

ΔKE = PE
½ mv2 = 2FTsinθ

We want to multiply the force of tension by two because the arrow bisects the string in two so that there are two equal tension forces acting on the arrow.  Additionally we only are concerned with the x component of the tension force because the y components of the two tension forces are equal but in opposite directions so that they cancel with each other.

Therefore:

½ (0.5 kg)(176 m/s)2 = 2FTsin(30˚)
FT = 7744 N = 7700 N