The position of a cart moving along a straight track is modeled by two different functional rules.
• For 0≤t≤3 seconds, s(t)=4t2 meters.
• For 3<t≤6 seconds, s(t)=36+8(t−3) meters.
The position function is continuous at t=3 s and differentiable everywhere except possibly at that point.
(a) Determine the velocity function v(t) for both time intervals.
(b) Determine the acceleration function a(t) for both time intervals.
(c) Sketch a fully labeled graph of the acceleration a(t) on 0≤t≤6, indicating any discontinuities.
(d) Identify all values of t in 0≤t≤6 at which the acceleration is not continuous, and explain the physical significance of each discontinuity.
(e) Compute the total distance traveled by the cart on 0≤t≤6 seconds.