Mathematics

Problem

A sprinter starts from rest at

t=0

and covers distance along a straight track such that the displacement function is

x(t)=100\,(1-e^{-kt})

metres, where

k

is a positive constant. a. (i) Derive an expression for the instantaneous velocity

v(t)

. b. (ii) Obtain the average velocity of the sprinter during the first

5\,\text{s}

in terms of

k

. c. (iii) Show that there is a time

t_{1}

(0<t_{1}<5\,\text{s})

for which the instantaneous velocity equals the average velocity found in part (ii), and find

t_{1}

in terms of

k