Projectile's Range There is a projectile launched from earth. It has an elliptical orbit unless it hits again earth. It has reached to apogee point then hit again Earth. I know initial, hit and Apogee point velocities. I calculated the time of projectile's duration of flight and launch angle. How can i find the range of projectile on Earth.
I searched a little bit. There are some answers to that but they assume Gravity is contast. In my problem gravity is not constant.
Given : $V_a$ = 1000 m/s
initial velocity and hit velocity $V_h$ = 4000 m/s
Found = Radius of apogee $r_a$
With conservation of angular momentum equation flight path angle
With $$r(\theta) = (h^2/Mu) \cdot 1/(1+e.\cos(\theta))$$
theta represents true anomalies, h angular momentum, Mu = Mass of Earth and Gravitational constant
I found $e$, eccentricity, with using $\theta = 180^{\circ}$.
then I found with same eq. theta initial and hit point true anomalies.
I calculated semi major axis length with the help of same equation using $\theta = 0$ point of apogee.
I calculated $T$, period
With using true anomalies I found eccentric anomaly then time between them.
 A: Some hints:
The orbit is an ellipse - and because it was launched from the surface of the earth, it WILL hit the earth again.
You have enough information to calculate the semimajor axis (using the vis viva equation). This axis goes through the center of the earth.
You know the initial velocity, and velocity at apogee. The gravitational potential follows a $\frac{1}{r}$ law. That is enough to get the maximum height of the projectile (from conservation of energy).
Now you have everything you need to draw the elliptical orbit, and the intersection of that orbit and the circle (centered on a focal point of the orbit)
UPDATE 
Since you found the anomalies, the following diagram should tell you trivially what the range is:

Obviously, $2\alpha + \phi = 2\pi$.
A: The. orbits have been sketched by Newton for different velocities at a given angle of projection. Quantities $p, \epsilon $ of classical ellipse $ p = r + \epsilon x $  with Earth center as one focus are expressed in terms of angle projected, projectile speed, mass of earth. Velocity less than escape velocity makes re-entry and re-hit with earth possible when no wind resistance is taken into computation.
