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jasonchiang97
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Homework Statement
A cannon that is capable of firing a shell at speed v0 is mounted on a vertical tower of height h that overlooks a level plain below.
Show that the elevation angle α at which the cannon must be set to achieve maximum range is given by the expression
csc2(α) = 2(1+gh/V02)
Homework Equations
x(t) = v0tcosα
y(t) = v0tsinα - (1/2)gt2 + h
The Attempt at a Solution
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First, I used
x(t) = v0tcosα
y(t) = v0tsinα - (1/2)gt2 + h
I solved for t in y(t) and plugged it into x(t) to get
x = [v0cosαsinα+v0cosα(v02sin2α+2gh)1/2]/g
then I solved for v0cosα(v02sin2α+2gh)1/2
and squared both sides to get
(xg - v02cosαsinα)/(v02cos2α) = v02+2gh
I expanded the left side of the equation and simplified with the right hand side to get
(g2/v02)x2 -(2gtanα)x - 2gh = 0
Solved for x to get
x = v02/2(sinα + √(sin2α +2h/v02))
I took the dx/dα = 0 to get the maximum angle
dx/dα = 1+sinα(sin2α +2h/v02)-1/2 = 0
And I said that it is max when sinα = 1 so then
xmax = v02/2 + v02/2(1+2h/v02
I'm unsure on how to obtain cscα from this
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