## rulnick.com/blog‣nfl‣190128k.txt

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2019-01-28 ~ 2019-01-29

### Field Goal Velocity and Angle

MIT Technology Review recently asked for the minimum initial velocity and associated vertical kick angle for a field goal attempt of horizontal distance $x$ to strike the crossbar at height $y$.
Let $v$ and $\theta$ be the initial velocity and angle, respectively. Then the horizontal and vertical travels of the ball at time $t$ after the kick are \eqalign { x &: t\mapsto vt\cos\theta, \cr y &: t\mapsto vt\sin\theta-\frac12 gt^2, \cr } respectively, where $g$ is the acceleration due to gravity. Solving the first equation for $t$, substituting into the second, solving for $v$, and zeroing the derivative with respect to $\theta$ produces the minimum velocity $v^*$ by way of the associated angle $\theta^*$: \eqalign { v^* &= \sqrt{g\left(y+\sqrt{x^2+y^2}\right)}, \cr \theta^* &= \frac{\pi-\arctan(x/y)}2. \cr } Here's what this looks like for typical NFL conditions:
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