The velocity of a particle moving along a straight line is modeled

MATHEMATICS · Multi-part · A Level · Mathematics · Kinematics · Kinematics

Problem

The velocity of a particle moving along a straight line is modeled, for

0\le t\le8

seconds, by the piecewise function

v(t)=\begin{cases} \dfrac{1}{2}t^2-2t+3,&0\le t\le4,\\ -\dfrac{1}{4}(t-4)^2+1,&4<t\le8. \end{cases}

Velocity is measured in meters per second and

t

in seconds. (1) Part 1: (a) Determine the acceleration

a(t)

for

0<t<8

, stating your work separately on each piece of the domain. (2) Part 2: (b) For

0<t<8

, find all times at which the acceleration is zero. Classify each zero of

a(t)

as corresponding to a local maximum or minimum of

v(t)

, and justify your classifications. (3) Part 3: (c) Compute the particle’s displacement and the total distance it travels during

0\le t\le8

seconds. Explain how sign changes of

v(t)

affect the total distance calculation. (4) Part 4: (d) On which subintervals of

0<t<8

is the particle’s speed increasing? Provide full justification based on the relationship between the signs of

v(t)

and

a(t)