Mathematics

Problem

For

t>0

, the velocity

v(t)

(in meters per second) of a particle moving along a horizontal line satisfies the implicit equation

t\,v(t)+\ln\bigl(v(t)\bigr)=t^{2}.

t=1

second, the particle’s velocity is

v(1)=1\,\text{m/s}

. a. Part 1: Derive an explicit expression for the particle’s acceleration

a(t)

in terms of

t

and

v(t)

. b. Part 2: Compute the acceleration at

t=1

second. c. Part 3: Determine whether the speed of the particle is increasing or decreasing at

t=1

. Justify your answer numerically and analytically.