How to Determine the Coefficients of a Gaussian Function from Two Known Points

I recently came across a problem in my research where I needed to calculate the form of a Gaussian function from two points that represent known solutions.  I wrote up my solution in the hopes that it will save other people the trouble of deriving it.  The link is here:

http://www.unc.edu/~haksaeng/mathematics/calculate_gaussian_curve.pdf

The backstory:

I am analysing the trajectory of some tennis balls that were thrown into the air due to an underground explosion.  I have decided to try modelling the force exerted on the tennis balls as a Gaussian pulse.  However, I don’t know exactly when the explosion started, or when the force is at its maximum.  All I know is the timing of two points:

1.  When the force equals the force of gravity (so the ball starts to move).

2.  When the force equals the force of gravity plus drag (when the explosion force becomes lower than gravity plus the air resistance caused by the ball flying through the air).

I don’t know the magnitude of the force except that it exceeds gravity at some point (obviously) and is greater than or equal to the force of gravity plus drag when the ball starts to slow down.

So my plan is to derive an expression for a Gaussian function based upon my two known points, and perform a nonlinear regression to figure out what magnitude of force gives the observed displacement of the tennis ball.

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About glossarch

The word "glossarch" doesn't exist. At least, not yet. But let's pretend it does for a second. The first part is "gloss," a word that comes to us from Ancient Greek via Latin and English. It means "language." The second part also comes from Ancient Greek and can mean "having power over." So "glossarch" means simply "language controller." So what am I doing making up words? Well, I made up an entire language once. It's called Angosey. So I'm the Glossarch of Angosey. I'm currently a doctorate student in volcano seismology (a branch of geophysics). I enjoy writing fiction and poetry, launching balloons, programming, and hanging out with my lovely wife! Follow me on Twitter! Writing and language creation: @glossarch Balloons and science: @bovineaerospace
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