How can we theoretically calculate the number of balloons needed to stop a bullet? After watching this video, I am interested in proving it mathematically. The problem is that how can we apply the conservation of momentum here to find the velocity of the bullet while we don't know the velocity for the exploded balloons after collision. Do I have to measure the velocities by setting up a real experiment?
 A: I'll have an attempt at answering this. Typically Balloons are made of latex, which has a Young's Modulus of $$ 0.1 \times 10^{9} GPa $$
Bullet weights 0.008kg (8g)
Radius of bullet is 9mm.
Speed of 759.968 m/s.
It takes a deformation of 2.5cm to break a latex balloon.
Using the equation for youngs modulus and calculating the work needed to pierce both sides of a balloon balloon with a bullet, we can determine how many are needed to stop it.
Young's modulus:
$$ Y_{m} = \frac{\frac{F}{A}}{\frac{x}{\triangle x}} $$
$$ F = \frac{Y_mAx}{\triangle x} $$
$$ A = \pi \times 0.009^{2} \hspace{0.2cm}\text{m} = 2.5 \times 10^{-4} \hspace{0.2cm} m^{2} $$
$$ x = 0.0001 \hspace{0.2cm}\text{m} $$
$$ \triangle x = 0.025\hspace{0.2cm}{m} $$
Using modulus formula, times 2 for each side of the balloon, we get how much force is needed to be applied to 0.00025 m squared of rubber to pierce both sides, which is
$$ F = 100 \hspace{0.2cm} \text{N} $$
so that's 200 Newtons (roughly) to get through one balloon, multiply by the deformation required to break a balloon, 0.025m, thats 5 Joules absorbed per balloon.
The 9mm bullet of has an energy of 2310J, meaning we need 462 balloons to stop a bullet.
