A particle starts moving rectilinearly at time `t=0` such that its velocity `v` changes with time `t` according to the equation `v=t^(2)-t`, where `t` is in seconds and `v` in ` s^(-1)`. Find the time interval for which the particle retards.

Explanation

Acceleration of the particle, `vec a=(d vec v)/(dt)=2t-1`
The particle retards when acceleration is oppostie to velocity. Hence, acceleration vector and velocity vector should be opposite to each other or the dot product of `vec a` and `vec v` should be negative.
`rArr vec a. vec v lt 0`
`rArr (2t-1)(t^(2)-t) lt 0`
`rArr t (2t-1)(t-1) lt 0`
`t` is always positive.
`because (2t-1)(t-1) lt 0`
`rArr` Either `2t-1 lt 0` or ` t-1 lt 0`
`rArr t lt (1)/(2) s` and `t lt 1 s`. This is not possible.
or `2t-1 lt 0` and `t-1 lt 0 rArr t g t (1)/(2)s` and `tlt 1 s`. Hencec, the required time interval is `(1)/(2)s g t t g t1 s`.