Projectile: Given V0, h, show d=(v0/g)sqrt((v0)^2-4gh)

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The discussion focuses on deriving the formula for the distance d between two points at height h for a projectile fired with initial velocity v0. The user initially formulates a quadratic equation and applies the quadratic formula, but struggles to simplify it to the desired expression. After several attempts and edits, they manage to manipulate their equation correctly, ultimately arriving at the required formula. The user expresses relief at solving the problem just in time before the deadline. The thread highlights the challenges of working with projectile motion equations and the importance of careful algebraic manipulation.
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Homework Statement


A projectile is fired from a gun (adjusted to give maximum range) with velocity v_{0}. The projectile passes through two points at a height h. The problem asks us to show that d=\frac{v_{0}}{g}\sqrt{v^{2}_{0}-4gh}
where d is the distance between the two points at height h.

Homework Equations


r=v_{0}t+\frac{1}{2}at^{2}
v=v_{0}+at
x= \frac{-b\pm\sqrt{b^{2}-4ac}}{2a}


The Attempt at a Solution


I was able to get a quadratic function of x:
0=\frac{g}{v^{2}_{0}}x^{2}-x+h

After manipulation using the quadratic formula, all I can see is:
x=\frac{v^{2}_{0}}{2g}+\frac{1}{v_{0}}\sqrt{v^{2}_{0}-4gh}

Which just looks so close but I'm killing myself in trying to see how it is either (1) wrong or (2) able to be simplified.

EDIT: x=\frac{v^{2}_{0}}{2g}+\frac{1}{2gv_{0}}\sqrt{v^{2}_{0}-4gh}, sorry.

Help?
 
Last edited:
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Does anyone even have any suggestions? This is actually due in about an hour and a half. I'm not heartbroken or anything but I'm feeling pretty annoyed that I might not get this problem. I honestly can't see what's going wrong here. Any creative suggestions or strong nudges are totally welcome.

Thanks...
 
Alright, so, I currently have the following written on my paper:

x= \frac{v^{2}_{0} \pm v_{0} \sqrt{v^{2}_{0}-4gh}}{2g}

I can't find the correction that makes this into the formula asked for.
 
Ahhhhhhh... So, this one was staring me in the face. Done and with 20 minutes to spare.
 
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