The problem facing every high school and introductory college physics student is that of a projectile fired at an angle ignoring air resistance. It’s fairly simple to derive from the first principles differential equations (with initial velocity , angle , and vertical acceleration . In this analysis, we wish to know what the cannoneers of old wanted to know: What is the range, or horizontal distance traveled before the cannon ball strikes the ground? We’ll call the range *R*, though in the following diagram it shows up as *x*.

We wish to know the range *R*, and we know and . Again, is dimensionless so dimensional analysis will tell us nothing about it. I don’t include the ball’s mass since we know that it will have no effect, ie. if we include it in the dimensional analysis it will have power 0 just like in the pendulum’s case. What’s the relevant physics we need to consider? There’s no air resistance, but there’s certainly gravity so we include *g*. Then we have

We could leave it at that, or we could consider the limiting or easy cases of to guess the angle dependence of the range. For , the ball leaves the cannon and then immediately strikes the ground so . For the ball flies straight up and then returns to the origin, so we again have . The first condition is satisfied by including, in the equation for range, a term and the second is correspondingly satisfied with a term. Therefore,

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The actual formula is

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