Question

A particle moves along a straight line and its position at time t is given by

Answer 1

(A) Use interval notation to indicate the time interval or union of time intervals when the particle is moving forward and backward

the particle is speeding up

if  a(t) >0, it means 12t-48>0, and t >4,  I = ]4, infinity[

the particle is speeding down

if  a(t) < 0, it means 12t-48>0, and t < 4,  I = ] -infinity, 4[

 

when the particle is moving forward and backward

that is depending of the sign of v(t)

moving forward if v(t)> 0

moving backward if v(t)<  0

but v(t) = 6t² - 48 t +90, this quadratic equation has exactly two solutions, t =1, and t =3

so  after taking care the sign of 6t² - 48 t +90,

 

v(t)> o when  t is in I = ] –inf, 1[ U ]4, inf[,the particle is moving forward

v(t) < o when  t is in I = ]1, 4[,the particle is moving backward