Win a Hat! (Its a really nice hat, its a really hard problem)

[John]

Ignoring the height change, it is an easy solution, and as several people have said, it is very easy to do yourselves. Chances are, your uncertainty in your velocity will make a bigger difference than the height variation (as a result, you are going to have to calibrate your shooter by testing it anyway, the equations will just give you a ballpark value to start with).

If you do try to account for the height change, I don't believe there is a closed form solution. I end up with a 6th order polynomial in sin(Theta),which does not have a general solution. You would have to numerically solve the equation for the appropriate values of x and v. Here is as far as I got:

The height at a given distance will be (measured from the release point):

y = x tan(Theta) - g * x^2 / (2 (v cos(Theta))^2) = x tan(Theta) - 32.2 * x^2 / (2 (v cos(Theta))^2)

This needs to match the height that you need to shoot above the release point. If I am understanding correctly:

y(x) = ht = 8.083 - 1.4381 sin(Theta) - 0.8147 = 7.2683 - 1.4381 sin(Theta)

For simplicity, define A = 7.2683 and B = 1.4381:

Combining the two equations:

x tan(Theta) - 32.2 * x^2 / (2 (v cos(Theta))^2) = A - B sin(Theta)

We could probably solve the preceding equation numerically for any particular values of x and v, but we will try to solve it symbolically instead.

Substitute a new variable, u = sin(Theta). This also implies that cos(Theta) = sqrt(1-u^2), and tan(Theta) = u/sqrt(1-u^2). If we solve for u, we have Theta = asin(u) :

x u/sqrt(1-u^2) - 32.2 * x^2 / (2 (v^2 (1-u^2))) = A - B u

Define C = 32.2 * x^2 / (2 v^2):

x u/sqrt(1-u^2) - C/(1-u^2) = A - B u

Move everything but the first term to one side:

x u/sqrt(1-u^2) = C/(1-u^2) + A - B u

Square both sides (easily done using WolframAlpha, etc.):

x^2 u^2 / (1-u^2) = ( (u^2-1)(A – B u) – C )^2 / (u^2-1)^2 = ( (u^2-1)(A – B u) – C )^2 / (1 - u^2)^2

Multiply both sides by (1-u^2)^2

x^2 u^2 * (1-u^2) = ( (u^2-1)(A – B u) – C )^2

Expand everything:

x^2 u^2 – x^2 u^4 = A^2 u^4-2 A^2 u^2+A^2-2 A B u^5+4 A B u^3-2 A B u-2 A C u^2+2 A C+B^2 u^6-2 B^2 u^4+B^2 u^2+2 B C u^3-2 B C u+C^2

Move everything to one side:

0 = A^2 u^4-2 A^2 u^2+A^2-2 A B u^5+4 A B u^3-2 A B u-2 A C u^2+2 A C+B^2 u^6-2 B^2 u^4+B^2 u^2+2 B C u^3-2 B C u+C^2 - x^2 u^2 + x^2 u^4


Collect terms:

0 = B^2 u^6-2 A B u^5-2 B^2 u^4 + x^2 u^4 +A^2 u^4+4 A B u^3+2 B C u^3-2 A^2 u^2 - x^2 u^2+B^2 u^2-2 A C u^2-2 B C u-2 A B u+A^2+2 A C+C^2

0 = B^2 u^6-2 A B u^5 + (x^2+A^2 -2B^2) u^4 + (4 A B +2 B C) u^3 +
(-2 A^2 - x^2 +B^2-2 AC) u^2 + (-2 B C u-2 A B) u+A^2+2 A C+C^2