What Angle of Projection Makes Maximum Height Equal to Range?

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The problem involves determining the angle of projection for a projectile fired at an initial velocity of 53 m/s, such that its maximum height equals its range. The relevant equations are R = Vo² * sin(2x) / g and H = Voy² / (2g), where g is the acceleration due to gravity (9.81 m/s²). By equating the maximum height (H) to the range (R), the derived formula is Voy² = 2 * Vo² * sin(2x). The challenge lies in isolating the angle of projection (x) from this equation.

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kuakkgom
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This problem has been giving me headaches for the past two weeks. Here it is, as well as how far I have gotten on it:

A projectile is fired with an initial velocity of 53m/s. Find the angle of projection such that the maximum height is equal to it's range.

So that basically boils down to:

H = R
Vo = 53m/s
ay = -9.81m/s^2
t = unknown
x (angle) = find

According to the nifty sheet the teacher tossed out the applicable formulas would be:

R=Vo^2*sin(2x)/g (since Ho and Hf will be the same)
H=Voy^2/2g.

Making H=R gets the formula Voy^2=2*Vo^2*sin(2x) ...which doesn't help me since x is what needs to be found, and without x one can't break Vo into component form.

Do I have the right idea or have I flubbed it already?

Thanks,

-K
 
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Voy=vo*sinx
 

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