I think the key item to note in the kinetic energy equation is that the energy is related to the SQUARE of the velocity. Which means doubling the speed quadruples (x4) the energy. In order to stop, the brakes have to convert that energy to heat (minus a little frictional loss here and there). Note in car tests what the stopping distances are at 60mph and at 70mph. A common figure at 60mph is 80ft, but at 70mph that often stretches to 140ft. See what a difference 10mph makes in stopping distance?
Also, distance=1/2*(acceleration in feet per second squared)*(time)^2. This assumes starting from rest. So if the acceleration rate is not constant, you can't solve for distance (as noted by smkettner)as there is one equation with two unknowns.
The other important equation is v^2 = 2*(acceleration)*(distance).
With those two equations you can figure out a lot, assuming constant acceleration.
Steve