[img]http://puu.sh/55VTj.png[/img] (edit: img tags don't work but you can click it)
okay, so for this question, i figured out that circle/ellipse/hyperbola involves trig functions when you parametrize it in terms of 't' i.e. x=cost y=sint for a circle, so when you differentiate it twice to get the acceleration function it will have a trig function which is non-constant since trig functions oscillate
okay, and for parabola, i can see that if y=x^2 then x=t and y=t^2 so y will be a constant acceleration when differentiated twice
but for a straight line i.e. y=x then you have x=t y=t and when you differentiate both 2 times you have a(t)= <0, 0> which is a zero acceleration. how can a straight line have non-constant non-zero acceleration?
Strictly speaking a "line in space" does NOT have "acceleration" at all! A "line in space" is a geometric concept, not physics.
You can, of course, write parametric equations for a straight line or curve in space, in terms of parameter "t" and then think of "t" as "time" and the line as the path of an object moving. (Personally, I dislike forcing physics terms on mathematics concepts like that.)
Then we can have "non-zero acceleration" just as a car moving down a straight road can speed up or slow down.