A particle moves along a straight line. Its velocity v (in meters per second) is described as a piecewise-linear function of time t (in seconds) as follows.
Velocity function v(t):
• For 0≤t≤3, v decreases linearly from 6 to 0.
• For 3<t≤5, v remains constant at 0.
• For 5<t≤9, v increases linearly from 0 to 8.
• For 9<t≤12, v increases linearly from 8 to 10.
• For t>12, v is constant at 10.
(a) Determine the acceleration a(t) for all t≥0. Give a piecewise description and include units in your answer.
(b) Identify every value of t at which a(t) is not defined. For each such value, state the one-sided limits of a(t) as t approaches that point and describe the graphical feature of v(t) that causes the non-differentiability.
(c) Sketch the graph of a(t) for 0≤t≤15. Label all intercepts and discontinuities clearly.
(d) The particle’s position function is s(t)=∫0tv(u)du with s(0)=0. Without performing any additional integration, determine on which subintervals of [0,15] the graph of s is concave up and on which it is concave down. Justify your answer in terms of the sign of a(t).