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Dustgil
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Homework Statement
A cannon that is capable of firing a shell at speed [tex]V_0[/tex] is mounted on a vertical tower of height h that overlooks a level plain below.
(a) Show that the elevation angle [tex]\alpha[/tex] at which the cannon must be set to achieve maximum range is given by the expression
[tex]csc^2(\alpha)=2(1+\frac {gh} {V_0^2})[/tex]
(b) What is the maximum range R of the cannon?
Homework Equations
[tex]F=m\frac {d^2r} {dt^2}[/tex]
The Attempt at a Solution
The only force on the particle is -mgk, so
[tex]m\frac{d^2r} {dt^2} = -mgk[/tex]
[tex]\frac{d^2r} {dt^2} = -gk[/tex]
[tex]\vec{v}=-gtk+v_0[/tex]
[tex]\vec{r}=\frac {-gt} {2}k+v_0t+r_0[/tex]
The initial velocity and position can be expressed as such
[tex]V_0=V_0cos(\alpha)i+V_0sin(\alpha)k[/tex]
[tex]r_0=hk[/tex]
Therefore vector r is
[tex]\vec{r}=V_0tcos(\alpha)i+(h+V_0tsin(\alpha)-\frac{gt^2} {2})k[/tex]
In component form,
[tex]x=V_0tcos(\alpha)[/tex]
[tex]z=(h+V_0tsin(\alpha)-\frac {gt^2} {2})k[/tex]
when z=0 (taking just the postitive root of the quadratic formula because we are concerned about positive t)
[tex]t= \frac {V_0sin(\alpha) + \sqrt{V_0^2sin^2(\alpha)+2gh}} {g}[/tex]
Plugging this into x, we get
[tex]x=V_0cos(\alpha)(\frac {V_0sin(\alpha)+\sqrt{V_0^2sin^2(\alpha)+2gh}} {g})[/tex]
This is the equation of motion in the x direction. It seems to make sense because if the angle is 90 degrees, we get 0 distance traveled. When the angle is 0 degrees, the y component of the initial velocity cancels out to give something nice. My thinking from this point is that taking dx/da and setting it equal to 0 would give the angle that would make x a maximum. However, when I do this I get a horribly complicated equation that I can't figure out how to simplify. Am I on the right track?