I sense a typo.Originally Posted by radyk05
Edit:
Try y = y<sub>0</sub> + v<sub>y,0</sub> * t - (1/2) * g * t²
Edit2:
While I'm at it:
v<sub>x</sub> = dx/dt = v<sub>x,0</sub>
v<sub>y</sub> = dy/dt = v<sub>y,0</sub> - g * t
At the max height reached, v<sub>y</sub> = 0, therefore t<sub>max height</sub> = v<sub>y,0</sub>/g
At that t<sub>max height</sub>, half the horizontal distance is covered, so x<sub>max height</sub> = x<sub>0</sub> + v<sub>x,0</sub> * v<sub>y,0</sub>/g = sin θ * v * cos θ * v /g, which is max for θ = pi/4.
Now we can clean the mess, and rewrite:
x = x<sub>0</sub> + sqrt(2)/2 * v * t
y = y<sub>0</sub> + sqrt(2)/2 * v * t - (1/2) * g * t²
Now, let's take a point y<sub>h</sub>.
y<sub>h</sub> = y<sub>0</sub> + sqrt(2)/2 * v * t<sub>h</sub> - (1/2) * g * t<sub>h</sub>²
h = y<sub>h</sub> - y<sub>0</sub>
h = sqrt(2)/2 * v * t<sub>h</sub> - (1/2) * g * t<sub>h</sub>²
We solve this for t<sub>h</sub>, we get:
t<sub>h,2</sub> = sqrt(2)/2g *(v + sqrt(v² - 4*h*g))
and
t<sub>h,1</sub> = sqrt(2)/2g *(v - sqrt(v² - 4*h*g))
d = x<sub>2</sub> - x<sub>1</sub> = sqrt(2)/2 * v * (t<sub>h,2</sub> - t<sub>h,1</sub>)
d = sqrt(2)/2 * v * (sqrt(2)/2g *(v + sqrt(v² - 4*h*g)) - sqrt(2)/2g *(v - sqrt(v² - 4*h*g)))
d = v/g * sqrt(v² -4*h*g)
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