case-1:
the motion or you can say the orbit of a projectile is parabolic because gravity acts on it.
case-2:
the motion or the orbit of an object thrown straight up ( at less than the escape velocity) or thrown straight down is a straight line
case-3:
then we have the planetary motion which is elliptical and of course it is because of the gravity
now I am going to the point,
the general formula of gravitational force is,
$$F=\frac{G\cdot m_1 \cdot m_2}{r^2}$$
now, how would you derive the equation of the parabola, straight line and ellipse from the general formula of gravity for the given cases.
I also want to know other types of orbit and the algebraic equation for that orbits motion.