Question

If a ball is rolling down an inclined ramp straight, will the change in position be squared for every passing second or would it just be multiplied by two? I'm not sure if this makes sense so here's what I mean in an example for the change in position will be squared for every passing second : 1 Second = 4 cm, 2 seconds = 16 cm, 3 seconds = 256 cm... etc

Answer 1

so
dx/dt = a t
dx = a t dt
at t = 0, dx = 0
at t = 1, dx = a
at t = 2, dx = 2a
at t = 3, dx = 3a
at t = 4, dx = 4a etc
in other words you are looking at v(t), the velocity, which is linear with time (the CHANGE in position, not the position)
v = a t

the POSITION itself will be quadratic in time of course. When you integrate x = k t you get x = (1/2)kt^2
POSITION is then

Answer 2

Thank you! And just to be sure I really understand what you're saying would this example be correct?
0 = 0
1 = 4 = a
2= 2(4) = 8
3 = 3(4) = 12
4 = 4(4) = 16
5 = 5(4) = 20

Answer 3

yes,your table is for a = dv/dt = 4
so dv = 4 * dt
dv/dt = a = 4
v = a t = 4 t
x = (1/2) a t^2
t a v x
0 4 0 0
1 4 4 2
2 4 8 8
3 4 12 18
4 4 16 32
5 4 20 50
It is that v column you want, how far it moves PER SECOND
not how far it moved total.