Proof: maximum range of an object launched off a cliff

AI Thread Summary
The discussion focuses on deriving the maximum range of an object launched off a cliff with height H, initial velocity v, and launch angle theta. The formula for the optimal angle is presented as cos^2(θ)=(v^2)/(2v^2+2gh). The poster struggles with the calculations, resulting in complex equations rather than the desired simple solution. They emphasize the need to first determine the total time of flight, which includes the ascent to maximum height and the descent back to ground level. The discussion highlights the challenges of applying quadratic equations in this context.
pantheid
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I've been working on this for a while, with little luck. We are told that an object is on a cliff of height H, and is launched off with a velocity v at an angle of theta with the vertical. gravitational acceleration is g. To get the maximum range, it must be a certain angle theta, given by the formula:
cos^2(θ)=(v^2)/(2v^2+2gh). Prove that this is correct.



v*sin=range/t, -h=v*cos*T-.5gT^2, getting a quadtratic theorem, and deriving. No luck, I get ugly giant equations rather than that small solution.
 
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First find out how much time is needed. Do that by looking at the vertical component. The total time is the time it takes to go up to its maximum height, plus the time it takes to fall back to the original height, plus the time it takes to reach the ground.
 

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