A particle begins at position meters on a coordinate line
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A particle begins at position s(0)=5 meters on a coordinate line. Its acceleration is a(t)=6t−12 meters per second squared for 0≤t≤5, and its velocity at time t=0 is v(0)=−2 meters per second.
(a) Derive explicit expressions for the velocity v(t) and position s(t) of the particle for 0≤t≤5.
(b) Determine all times in the interval 0≤t≤5 at which the particle is at rest. Classify each as a change in direction or a momentary stop without reversal. Justify your classification.
(c) Calculate the particle’s net displacement over 0≤t≤5.
(d) Find the total distance traveled on the same interval and state it to the nearest hundredth of a meter.
(e) Suppose the interval of observation is extended beyond t=5. Explain how the unbounded behavior of a(t) would influence the long-term relationship between total distance traveled and net displacement.