I have read that projectile motion is parabolic in nature. I know that the equation of projectile motion represents a parabola. But when do we get a parabolic curve? Does it happen when there is downward acceleration and horizontal velocity? What condition results in a parabolic shape?
Is there any possibility of elliptical motion or any other curve? What would be the determining factors in that case?
I am not sure how much you can generalize the geometry of motion other than to say all regular orbits are conic sections, and that parabolas are just a special case that arises when the direction of gravity is fixed (or the focus point is at infinity).
There are a lot of resources out there discussing the topic of the geometry of orbits if you explore more.
Imagine throwing a ball at some speed say ($v_0$). As you stated, the ball will follow a parabolic trajectory before hitting the surface.
Now, try doing the same with even greater speed ($v \gt v_0$) and repeat it again. There will be a point where the ball is fast enough to not come back to the surface. What do you think the shape of motion is in this case?
Well, it is an elliptical orbit.
Even more complex patterns and shapes can form. For eg: 2-body problem or the general n-body problem.
Well there is no general answer to the question. You find the path of particle(s) for specific cases and match it with the existing equations of curves, if it matches then you get the desired curve by the motion of the particle.
You get a parabolic curve if you throw in any direction not vertical in a homogeneous field so g=const as long as you consider the earth as a plain, so the speed should not be too high. Also it excludes friction. you can tie the ball at a string to get circular motion or on a spring or rubber band to get more complicated curves. Also if you reach the speed of a rocket you can have ellipse around the earth.
