## Friday, November 25, 2016

### Mike Mitchell's hit on T.Y. Hilton

Thanksgiving is a storied American tradition that has evolved into just as much of a football-viewing holiday as it is one to give thanks and enjoy incredible food. This year, the Indianapolis Colts hosted the Pittsburgh Steelers in one of the holiday's three big games.

Colts backup quarterback Scott Tolzien unleashed a downfield pass to wide receiver T.Y. Hilton, who performed a spectacular catch right before the endzone in order to secure the pass. Within fractions of a second, however, Steelers safety Mike Mitchell delivered a terrifying hit to Hilton in order to make the tackle:

To understand why both Mitchell and Hilton were removed from play with fears of injuries, let's model the collision as a 1-D elastic collision for the sake of simplicity.

• mass of Hilton = 82 kg
• mass of Mitchell = 95 kg
• v1 (velocity of Hilton) = 9 m/s
• v2 (velocity of Mitchell) = 4 m/s
We know from elastic collisions that both momentum and energy are conserved; these relationships may be combined into a simple expression that relates both the initial and final velocities of both entities involved in the collision--

v1 + v1' = v2 + v2'

...where v1' and v2' represent the post-collision velocities of Hilton and Mitchell, respectively. Solving this expression for v1' and then plugging the result into the expression for conservation of momentum allows us to solve for v2'.

v1' = v2 + v2' - v1

m1v1 + m2v2 = m1v1' + m2v2'

m1v+ m2v2 = m1(v2 + v2' - v1) + m2v2'

v2' = (2m1v1 + m2v2 - m1v2) / m1 + m2

With the proper values for Hilton and Mitchell substituted into the equation, the post-impact velocity of Mitchell (v2') = 8.6 m/s.

Plugging in the found value of v2' into the expression for v1' reveals that the post-impact velocity of Hilton (v1') = 3.6 m/s.

Using the expression for impulse, we can calculate the net acceleration that each player experiences, assuming that the collision lasted for approximately 0.03 s:

ΣF = (Δp) / (Δt)
ma = (m(vf - vo)) / (Δt)

Hilton experiences a net acceleration of 180 m/s^2, or 18.4 g's; Mitchell experiences a net acceleration of 153 m/s^2, or 15.6 g's.

These players are experiencing tremendous magnitudes of acceleration in an extremely short period of time-- it is pretty easy to see how injuries are such a big fear in the NFL.