When dealing with higher time derivatives like jerk, how does one find the distance traveled? Can it be calculated by just knowing time?
Knowing only "jerk" (third derivative of position), you cannot determine the distance traveled.
To get distance traveled (or equivalently, position as a function of time) from jerk, you need to integrate three times. Each integration produces a constant of integration representing an initial value; your final equation looks something like this:
$$p(t) = \iiint j(t) + at^2 + vt + x$$
where "a" is your initial acceleration, "v" is your initial velocity, and "x" is your initial position. "x" doesn't matter for computing distance traveled, but the other two do.
Integrate the jerk 3 times then using starting conditions to work out the integration constants.