[Phys-L] Re: projectiles

[David_Bowman at GEORGETOWNCOLLEGE.EDU (David Bowman)]

Regarding Anthony's comment:

> I actually tried an initial height of 1 m and initial speed of
> 1 m/s. Did a spreadsheet and got maximum range around 25°. Then I
> realized that the formula, if one exists, had to be very
> complicated.

The general formula result is not all that complicated, but the
intermediate algebra might be a little intimidating.  The problem is
a simple calculus minimization problem in 1 variable.

The formula for the horizontal range R in terms of the initial launch
speed v, the gravitational field strength g, the launch angle q, and
the launch height h is:

R = (v^2/g)*cos(q)*sin(q)*(1 + sqrt(1 + 2*g*h/(v*sin(q))^2))

The angle that maximizes R is:

q = (1/2)*arccos(1/(1 + v^2/(g*h)))

and the maximum range at this angle is:

R_max = sqrt(1 + 2*g*h/v^2)*(v^2)/g

David Bowman