Quarterback Safety in American Football By Len Huang and Megha Jain

QB Safety 1/23 Nice to Meet You!

Len Huang Megha Jain SCS ‘23 SCS ‘23

QB Safety 2/23 Agenda

● Intro to American Football ● Motivation ● Approaches to Modelling ● Proofs and Insights

QB Safety 3/23 Intro to American Football

● Line of Scrimmage ● Quarterback ● Offensive Linemen ● Defensive Linemen ● Wide Receiver ● Linebacker

QB Safety 4/23 Intro to American Football

Game of Advancing the Ball Down the Field. Played in Increments QB Safety 5/23 Intro to American Football (Line of Scrimmage)

Current Lengthwise Position of the Ball QB Safety 6/23 Intro to American Football (Quarterback)

Quarterback: Responsible for Passing the Ball QB Safety 7/23 Intro to American Football (Linemen)

Offensive Line Protects QB, Defensive Line Attacks QB QB Safety 8/23 Intro to American Football (Wide Receivers / Linebackers)

Wide Receivers Run and Catch, Linebackers Prevent This QB Safety 9/23 Motivation ● Safety ⇒ QB is not tackled ● Efficiency ⇒ Pass is made

● Stepping stone to more multi-agent systems

QB Safety 10/23 Approaches to Modelling

● Simplifications ● Modeling Scenarios ○ Linemen Collision (Safety) ○ Passing (Efficiency) ● KeyMaera X Model

QB Safety 11/23 Simplifications

Less Players + One Dimensional

QB Safety 12/23 Lineman Collision

QB Safety 13/23 Lineman Collision (Visualization)

QB Safety 14/23 Pre-collision: run Perfectly inelastic Post-collision: move with towards each other collision dampened speed towards QB

Lineman Collision (Movement) QB Safety 15/23 Lineman Collision (Conservation of Momentum) QB Safety 16/23 Passing

QB Safety 17/23 Passing (Passing ↔ Open)

● Efficiency ⇒ Pass is made ● Assume Quarterback can instantaneously pass as soon as our Wide Receiver is open ● Lemma: isOpen(...) == ballPassed(...) ● Why? ○ Focus on QB Safety ○ Helps us define “pass is made”

QB Safety 18/23 Passing (Defensive Considerations)

Man: Wide Receiver must run past the Linebacker in Zone: Wide Receiver must catch before getting order to catch the ball tackled by the Linebacker

Bool isOpen(WR,LB) ↔ LB < WR Bool isOpen(WR,LB) ↔ WR < LB

QB Safety 19/23 Keymaera X Model

● Variation ● Pre-collision Movement ● Post-collision Movement ● isOpen() == ballPassed() ● QB Unhurt + Ball is Passed

QB Safety 20/23 Proofs and Insights

● Tactic ● Finding the Time

QB Safety 21/23 Proofs and Insights QB Safety 22/23 Questions?

QB Safety 23/23