A particle moves along a line such that its acceleration
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A particle moves along a line such that its acceleration a(t), in meters per second squared, is given for 0≤t≤6 seconds by the following description.
• a(t)=3 for 0≤t≤2.
• On 2<t≤4,a(t) decreases linearly from 3 to −1.
• a(t)=−1 for 4<t≤6.
At time t=0, the particle’s velocity is 4 m/s and its position is x=0.
(a) Sketch a carefully labeled graph of the velocity function v(t) for 0≤t≤6. Indicate the exact coordinates of all corner points.
(b) Calculate the particle’s velocity at t=6 s.
(c) Determine the maximum velocity attained on 0≤t≤6 and the time at which it occurs. Justify your answer.
(d) Find the particle’s position at t=6 s. Show all steps in your reasoning.
(e) Explain how the concavity of the particle’s position–time graph on each subinterval relates to the sign of the acceleration given.