A sprinter runs along a straight track and follows the acceleration profile a(t)=…

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

Problem

A sprinter runs along a straight track and follows the acceleration profile

a(t)=\begin{cases} 4, & 0\le t<2,\\ k(t-2), & 2\le t\le5,\\ -3, & 5< t\le7, \end{cases}

where

a

is in metres per second squared and

t

is in seconds. At

t=0

the sprinter starts from rest at the origin. The design goal is for the sprinter to come to rest again (momentarily) at

t=5

seconds before accelerating in the negative direction. (a) Determine the value of the constant

k

that satisfies the design goal stated above. Show all reasoning. (b) With this value of

k

, derive a complete piecewise expression for the velocity

v(t)

0\le t\le7

seconds, making sure to handle continuity conditions at the transition times. (c) Establish an expression for the position function

s(t)

over the same interval, choosing the constant of integration so that

s(0)=0

. (d) Compute the sprinter’s position at

t=7

seconds, and interpret the sign of this value in the context of the motion. (e) Determine the time in

0\le t\le7

at which the sprinter is farthest from the origin. Justify your answer using appropriate calculus techniques.