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So I've been trying to work out how to reverse a formula I found and worked on. I have thoroughly tested these formulas at desmos.com/calculator and have a realistic physics simulation playing in real-time over and over. It works EXACTLY as I need it to, however, I noticed that given an Initial Velocity, Gravity, and Launch Height, I can calculate the maximum distance of the projectile by first finding the angle (theta) that produces the longest shot from height y0 using this formula , then using this formula to calculate the maximum range of the projectile.

I was wondering how I could reverse this so that given a Range, Gravity, and Height, you can find the Initial Velocity needed to reach it. I tried for about 3 hours to do it myself, manipulating the formulas and embedding the angle formula into the distance one and then simplifying it a LOT, but got stuck right at the end.

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That maximum range formula doesn't make sense. If it's being fired at an elevation and velocity greater than 0, and the angle is 0, according to the formula, the projectile would have a maximum range of 0. Try finding a different formula.

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