If you are having problems figuring out what equation to solve take a look at the following.
First I want to know the path the shell will follow through space. It's like putting a bright light on the shell, and photographing the entire flight with the camera's shutter stuck open. When I develop the film, I see a portion of a parabola that begins at the cannon and ends where the shell hits the ground.
I start with the basic laws of motion in the x and y direction.
x = v0* cos(angrad) * t (1) y = v0 * sin(angrad) * t - 0.5 * g * t**2 (2)v0 is 300 meters per second (about sound speed).
I solve Equation (1) for t and get:
t = x / (v0*cos(angrad))Now I substitute this into Equation (2) and get:
y(x) = x*tan(angrad) - (0.5*g*x**2)/(v0*cos(angrad))**2This is the equation for the light streak in our picture.
I'll need the derivative of this function later for Newton's method.
dy g*x -- = tan(angrad) - ------------------- dx (v0*cos(angrad))**2
Notice that I only need the derivative with respect to x. For any given solution "angrad" is treated as a known constant.
Now I have the trajectory of the shell. Next I have to find the point where the shell's trajectory intersects the ground. Mathematically that can be written as:
Now if I define a new function f(x) to be:
then I'm back to solving the problem f(x)=0, that we discussed in class, I've worked this out for you in newton1.f & newton2.f.
You want to solve f(x) = y(x) - h(x) = 0
You have y(x) above, and a specific expression for h(x)
All you need is the derivative of f(x)
df dy dh -- = -- - -- dx dx dx
Plug the new formulae for f(x) and dfdx into the right spots of my examples and you should be home free.