Finding the optimal launch for a projectile?

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To determine the optimal launch angle for a water balloon from a height H with an initial velocity Vo, Newton's Second Law can be applied to derive an equation for maximum horizontal distance (xmax) at an angle theta. The discussion highlights the need to create a second equation from xmax to find the optimal angle for maximum distance. The user encountered issues with their kinematic calculations, leading to an incorrect optimal angle of 89 degrees. Assistance is sought for correcting the derivation and applying root-finding methods like fzero in MATLAB to identify the true optimal angle. Proper calculations and methods are essential for accurately determining the projectile's trajectory.
Boatman(J)
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Suppose I wanted to launch a water balloon off a cliff of a known height of H and known initial Velocity of Vo, where we can neglect air resistance. Using Newton's Second Law and differentiating it, how can i derive an equation for maximum distance traveled (xmax) at an angle theta? Then from the equation derived for xmax, how can I create another equation that can be use to derive the optimal angle which allows the balloon to travel the xmax? I will then use that equation for the optimal angle to find the optimal angle by root finding with the use of fzero in matlab.

I did this through kinematics but it was done wrong and kept telling me that optimal angle was 89 degress.

Can someone please help?
 
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