A ball is launched vertically upward from ground level
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A ball is launched vertically upward from ground level. Its height above the ground, in meters, for 0≤t<8 seconds is modeled by the twice–differentiable function s given by
s(t)=40t−5t2−64t4.
Assume this model is valid until the ball returns to the ground at t=8 seconds.
(a) Show that t=0 and t=8 are not interior critical points for s on the interval (0,8), even though s(0)=s(8)=0. Provide a clear analytic explanation.
(b) Determine all critical numbers of s in the open interval (0,8) and present complete calculations.
(c) For each critical number found in part (b), decide whether it corresponds to the moment when the ball reaches a local maximum height, a local minimum height, or neither. Supply thorough justification for each conclusion.
(d) Identify the time at which the ball attains its greatest height during 0≤t≤8, state that maximum height, and explain how your earlier work supports this determination.
(e) Interpret the sign of s′′(t) at the time of maximum height in the physical context of the ball's motion.